At the same time, the Nazi regime was conducting another secret killing programme. Under the code name Aktion T4, around seventy thousand Germans with mental or physical disabilities were murdered in specialised killing centres using poison gas. The victims included thousands of institutionalised Jews. Although Hitler ordered a halt to the centralised programme in August 1941 after news leaked out, the killing continued in decentralised forms, and the personnel and techniques developed for T4 would be directly transferred to the murder of Jews. The gas chamber was not invented for the Holocaust, but it was perfected there.

On the twenty-second of June 1941, Germany invaded the Soviet Union. Hitler framed the war as an apocalyptic struggle against what he called Jewish Bolshevism, and ordered it conducted as a war of extermination, with no regard for the laws of war. The Einsatzgruppen, reorganised before the invasion for mass killings, followed the army eastward. They were initially ordered to shoot Soviet officials and Jewish state employees. But through the summer of 1941, as Heinrich Himmler, the head of the SS, visited the killing zones personally and relayed orders to expand the scope of murder, the killings escalated. By July and August, Jewish women and children were being shot alongside men. By the end of 1941, more than eighty per cent of the Jews in Latvia, Lithuania, and much of Ukraine and Belarus had been murdered.

The executions followed a pattern. Victims were rounded up without warning, marched to a site a few kilometres from a town, forced to undress, and shot into pits they had often been made to dig themselves. The preferred method was a single shot to the back of the neck. In the chaos, many were buried alive. The pits were sometimes guarded afterwards; sometimes a handful of the living crawled out. These were not hidden operations conducted in secret facilities. They were public spectacles, and the victims' possessions were looted by soldiers and local inhabitants alike. Executions peaked at forty thousand a month in Lithuania in the late summer, and reached their height in Belarus in the autumn.

The scale of the shootings quickly outpaced the capacity of the Einsatzgruppen, which numbered only around three thousand men. Himmler mobilised twenty-one additional battalions of Order Police to assist them. Regular army soldiers, Waffen-SS brigades, and local auxiliaries from across occupied Eastern Europe all took part. Romania killed the largest number of Jews after Germany itself, with Romanian soldiers murdering tens of thousands from Odessa alone by the spring of 1942, and deporting between a hundred and fifty thousand and a hundred and seventy thousand more from Bessarabia and Bukovina to ghettos in Transnistria, where the death toll may have reached a hundred and sixty thousand. The Holocaust was not carried out by Germany alone.

By December 1941, the decision had hardened into policy. On the twelfth of that month, Hitler told senior Nazi officials, invoking a prophecy he had made in 1939: 'The world war is here; the annihilation of the Jews must be the necessary consequence.' On the twentieth of January 1942, Reinhard Heydrich, head of the Reich Main Security Office, convened a meeting of senior officials at a villa on the Wannsee lake near Berlin. The Wannsee Conference, as it became known, was not where the decision to murder the Jews was made. It was where the logistics of murdering every Jew in Europe were coordinated across government departments. Heydrich presented a list of eleven million Jews across the continent. Every one of them was to be killed.

The instrument chosen for the majority of these killings was the extermination camp. The first was Chełmno, in the occupied Polish territory called the Wartheland, which began operations in December 1941 using gas vans: trucks into which victims were forced and killed by carbon monoxide from the engine exhaust. Then came the purpose-built killing centres of Operation Reinhard: Belzec, Sobibor, and Treblinka, all located in occupied Poland, all built on railway lines to receive deportation trains from across the continent. At Auschwitz, in East Upper Silesia, experiments on Soviet prisoners of war in late 1941 had established Zyklon B as the preferred killing agent, and by early 1942 it became the standard method in the extermination camps.

The process at these camps was designed for speed. Deportees arrived in overcrowded cattle wagons, sometimes after journeys of several days, with no food or water. Upon arrival, they were robbed of their remaining possessions, forced to undress, and their hair was cut. Then they were driven into the gas chambers. Death from the gas was agonising and could take up to thirty minutes. At Treblinka, Sobibor, and Belzec, virtually everyone on a transport was killed on arrival. At Auschwitz, around twenty to twenty-five per cent were selected for forced labour, though many of those died later of starvation, disease, or medical experiments. The camps required relatively few German guards; the bulk of the operational work was carried out by a small number of Jewish prisoner-workers who were themselves killed and replaced.

The murder rate was most intense during Operation Reinhard, from March 1942 to November 1943. Historians estimate that around one point four seven million Jews were murdered in just one hundred days between late July and early November 1942, a rate of killing that exceeds even the Rwandan genocide. Over three million Jews were killed in 1942 alone. On the third of November 1943, around eighteen thousand four hundred Jews were murdered at Majdanek in a single day as part of Operation Harvest Festival, the largest single massacre of Jews by German forces, in which some forty-three thousand people were killed over two days. The majority of Holocaust victims died in that single year of 1942.

While the extermination camps consumed their transports, the ghettos of occupied Poland were being emptied. The Warsaw Ghetto clearance began on the twenty-second of July 1942. Over the following weeks, two hundred and fifty thousand of the ghetto's three hundred and fifty thousand inhabitants were deported to Treblinka and gassed. A further ten thousand were shot inside the ghetto. Only around thirty-five thousand were left behind in a final selection, with another twenty or twenty-five thousand managing to hide. Misdirection had convinced many Jews that they could avoid deportation until it was too late.

When the surviving inhabitants of the Warsaw Ghetto finally understood that deportation meant death, they chose to fight. In April 1943, with a handful of pistols, rifles, and homemade bombs, Jewish fighters launched an uprising against the German forces sent to liquidate what remained. The Warsaw Ghetto Uprising lasted nearly a month before German forces, using artillery and setting the ghetto systematically on fire, crushed it. The fighters knew they could not win. They fought because they refused to be taken without resistance. The uprising prompted the Nazi leadership to accelerate the liquidation of remaining ghettos across occupied Poland, fearful of further resistance spreading.

Deportations were not confined to Poland. In France, Belgium, and the Netherlands, local police forces were used to arrest Jews for deportation, limiting the number of German personnel required. Nearly a hundred thousand Jews were deported from those three countries in 1942 alone. The death rate in the Netherlands was particularly high, which scholars have attributed to difficulties in hiding and a higher degree of collaboration among Dutch police. In Slovakia, the government handed over around fifty-eight thousand Jews to German custody between March and October 1942. Bulgaria cooperated in deporting eleven thousand Jews from territories it occupied in Greece and Yugoslavia to Treblinka, though it declined to deport Jews from its own pre-war territory, due to widespread opposition from within Bulgarian society.

Until the German occupation of Hungary in March 1944, the Hungarian government had largely protected its Jewish population. But once Germany moved in, the Hungarian authorities cooperated with devastating speed. Between March 1944 and the ninth of July 1944, four hundred and thirty-four thousand of Hungary's remaining eight hundred and twenty-five thousand Jews were deported on trains, mostly to Auschwitz, where the great majority were murdered immediately upon arrival. It was the largest and fastest deportation of the Holocaust's final phase.

Not all Jews were killed immediately. From 1938 onward, and with increasing intensity after 1941, hundreds of thousands were drafted into forced-labour camps and work details. Jewish forced labourers were typically assigned to unskilled, exhausting work and treated far worse than non-Jewish prisoners. Starvation, disease, abuse, and deliberate overwork killed enormous numbers. Private companies, including major German firms, used Jewish forced labour under murderous conditions and profited from it. Including Soviet and Polish civilians, around thirteen million people in total were brought to Germany for forced labour, and hundreds of thousands died.

In mid-1943, Himmler moved to bring surviving Jewish forced labourers under direct SS control within the concentration camp system. Some ghettos were redesignated as concentration camps; others were dissolved and their surviving inmates sent on. Despite everything, as many as two hundred thousand Jews survived the war inside the concentration camp system. They were the lucky fraction of a fraction. The image of the concentration camp, with its guard towers and striped uniforms, has become the most recognised symbol of the Holocaust, even though most Holocaust victims never entered one: they were shot in ravines or gassed in purpose-built death facilities.

Resistance took many forms. In the forests of Belarus and Ukraine, an estimated twenty to twenty-five thousand Jews joined Soviet partisan units. Thousands more lived as non-combatants in hidden forest camps. Inside the extermination camps themselves, Jewish prisoner-workers staged uprisings at Treblinka and Sobibor in 1943, forcing those camps to shut down earlier than planned. These acts of resistance were undertaken by people who knew they were almost certainly going to die; they fought because the alternative was to go without a word.

By the summer of 1944, the war was turning against Germany. Soviet forces advancing through occupied Poland liberated Majdanek in July 1944, and the evidence of industrial mass murder found there was made public. In November 1944, the last mass gassing at Auschwitz took place. The gas chambers were shut down and the crematoria were demolished in an attempt to hide the evidence. In January 1945, most of the remaining sixty-seven thousand Auschwitz prisoners were forced onto a death march westward into Germany.

The death marches were killing operations in all but name. Prisoners weakened by years of starvation and forced labour were driven through winter conditions with no food. Those who could not keep pace were shot at the roadside. Sick prisoners were often killed before the marches even began. Between October 1944 and the end of the war, around seventy thousand to a hundred thousand Jews died in these marches. On the twenty-seventh of January 1945, Soviet troops entered Auschwitz and found a few thousand survivors, the rest having already been evacuated westward. The liberators of camps across Germany in the following months found piles of corpses that had to be bulldozed into mass graves, and living prisoners so emaciated they were barely recognisable as human beings.

When the killing finally ended with Germany's defeat in May 1945, the scale of what had been done could begin to be measured, though it could never truly be comprehended. Around six million Jews had been murdered. Half of them had come from Poland. Around two point six million had lived within the 1941 Soviet borders. Jewish children and the elderly had died at even higher rates than adults. One third of the entire Jewish population of the world was gone. In some regions, the death rate approached one hundred per cent.

About half of the six million had been killed by poison gas. Around one point five to two million had been shot. The rest had died of starvation, disease, exhaustion, and deliberate abuse in ghettos, camps, and on the marches. Non-Jewish victims of Nazi persecution added millions more: Soviet prisoners of war died at a rate of sixty per cent in German custody, around three point three million people. Millions of Polish and Soviet civilians were killed. Hundreds of thousands of Roma were murdered. The total death toll of Nazi persecution, including all targeted groups, may have reached seventeen million.

A few perpetrators faced justice. In 1945 and 1946, the International Military Tribunal at Nuremberg tried twenty-three senior Nazi leaders. The United States Army alone charged over sixteen hundred defendants in hundreds of further war crimes trials. West Germany later investigated a hundred thousand people and tried more than six thousand, mainly low-level perpetrators. Adolf Eichmann, one of the chief organisers of the deportations, was captured by Israeli agents, tried in Jerusalem in 1961, and executed. Most perpetrators were never tried at all. An estimated two hundred thousand to two hundred and fifty thousand Germans had been directly involved in the killing, and if those who organised and enabled it are counted, the figure rises to half a million.

After liberation, many Jewish survivors attempted to return home. They found their property taken, their neighbours unwilling to return what had been stolen, and in some places active violence. A pogrom against Jewish survivors in Kielce, Poland, in 1946 killed dozens of people who had survived the entire war. Antisemitism had not died with the Nazi regime. Many survivors left eastern Europe entirely. The establishment of the State of Israel in 1948 gave a destination to hundreds of thousands who could not or would not return. Others eventually reached the United States when immigration restrictions were loosened around 1950.

Between 1945 and 2018, Germany paid around eighty-seven billion dollars in restitution and compensation to survivors and their heirs, a figure that falls far short of the losses suffered. Historians estimate that property losses to Jews in the major occupied and allied countries amounted to the equivalent of around a hundred and eighty billion dollars in today's terms, not counting the value of the labour extracted from forced workers. Poland, the country where more than half the Jewish victims had lived, remains the only member of the European Union that has never passed any restitution legislation.

International Holocaust Remembrance Day is marked each year on the twenty-seventh of January, the date Soviet forces entered Auschwitz. Memorials, museums, and educational programmes exist across dozens of countries. Holocaust denial is a criminal offence in several European nations. The scholarly literature on the Holocaust is vast, encompassing thousands of books. Historian Omer Bartov has described it as unique in being 'the industrial killing of millions of human beings in factories of death, ordered by a modern state, organised by a conscientious bureaucracy, and supported by a law-abiding, patriotic, civilised society.' That last word is the one that refuses to let go.

The Jewish population worldwide has still not recovered to its pre-war level. In 1939, there were around sixteen point six million Jews in the world. By the end of 2020, that number had reached fifteen point two million. The gap of one point four million represents not just those murdered but all the children and grandchildren who were never born because their parents and grandparents were killed before they could have them. The Holocaust did not end in 1945. Its arithmetic continues to compound, quietly, in the absence of people who were never allowed to exist.

What the Holocaust demands of those who learn about it is not simply grief, though grief is appropriate. It demands an understanding of how ordinary institutions, ordinary bureaucracies, ordinary soldiers, and ordinary citizens can be made to participate in the murder of millions, when the ideology that drives them is allowed to take hold without resistance. It was not carried out by monsters from outside history. It was carried out by people inside it, in the middle of the twentieth century, in one of the most educated and culturally sophisticated nations in Europe. That is the fact that makes it impossible to look away, and impossible to forget.