In 1955, 10 years after his liberation from Auschwitz, Primo Levi published an anguished article about the “gigantic death-dealing machine” the Nazis had built to wipe out Jews such as himself. Levi was mainly concerned with the 1950s, however, not the recent past. He feared that the greatest crime imaginable, still so vivid in the minds of survivors, was in danger of being forgotten by the wider public. Levi railed against the “silence of the civilised world”, which regarded any mention of Nazi extermination camps as in bad taste.

How things have changed. Far from being forgotten, the murder of European Jewry has become a global benchmark for judging inhumanity. Levi’s own memoir of Auschwitz, If This Is a Man, which was initially met with indifference, has been recognised as one of the “truly necessary books” (Philip Roth), and every year sees a stream of works by survivors and historians, philosophers and novelists. The question is no longer: “Is this silence justified?”, as Levi asked rhetorically back in 1955. It is now: “Which of the countless studies should we read?”

Laurence Rees’s The Holocaust: A New History is puffed on its inside cover as “the first accessible and authoritative account of the Holocaust in more than three decades”. Such PR bluster does the book no favours. For there really is no shortage of important recent works, among them Saul Friedländer’s unsurpassed survey The Years of Extermination, which won the Pulitzer prize; Timothy Snyder’s bold reinterpretation Black Earth; and the late David Cesarani’s deeply researched and highly readable study Final Solution, which appeared only last year.

Yet look beyond the hype and you will find a fine book. Rees skilfully charts the development of Nazi antisemitic measures, from segregation and discrimination in the years before the second world war to ghettoisation and deportation during the war, culminating in the systematic extermination of around six million Jews. As a former head of BBC TV history programmes, Rees has produced some of the most thoughtful documentaries about Nazi Germany, and he tells this story through a mix of closeups and wide shots of the historical landscape. The latter provides crucial context, connecting anti-Jewish policy to the changing fortunes of the war and the dynamics of Nazi rule. Rees also explains how the perpetrators learned from earlier attacks on other “community aliens”, such as the disabled – the first victims of mass gassings in the Third Reich, who were murdered in “euthanasia” centres equipped with fake showers.

Uprising leader Marek Edelman pays tribute at the Memorial to the Warsaw Ghetto Heroes. Photograph: Pawel Kula/EPA

The author tackles some persistent myths along the way. He shows that there was plenty of defiance by the victims, contrary to postwar claims that they followed their killers “like sheep”. Some even got their hands on weapons and turned them against Germans. One example is Marek Edelman, who fought in the doomed Warsaw ghetto uprising. “The first few days were our victory,” he recalled. “We were used to being the ones who ran away from the Germans. They had no expectation of Jews fighting like that.”

Rees also shows that the Nazi machinery of mass murder was far from impersonal or antiseptic, highlighting the sadistic violence of some killers, the carnage in death camps overflowing with corpses, and the unspeakable suffering of the doomed: when children were dragged away from their parents in the Łódź ghetto, one survivor remembered, “their screams reached the sky”.

At the centre stands the question of how the Holocaust happened. For Rees, there was nothing inevitable about it. While Hitler was an exceptionally vicious antisemite from the start of his political career – calling for the “uncompromising removal” of Jews as early as 1919 – he had no master plan for murder. Nazi policy followed a twisted road, with plenty of stops, starts and turns, before heading towards mass extermination. And even then, there was no single order from the top. Hitler set the direction, to be sure, but he left his underlings to devise ever more extreme measures to realise his vision. There were great variations across Europe, depending on local circumstances, leading the Nazis to implement the Holocaust “in radically different ways in different countries”.

These are not novel conclusions; they reflect the current historical consensus. In that sense, there is little new about this “new history”. What distinguishes it is not an original interpretation, but its approach. Rees is a gifted educator, who can tell a complex story with compassion and clarity, without sacrificing all nuances. It is this quality that makes his book one of the best general introductions to the Holocaust.

Above all, it comes alive through the voices of victims, killers and bystanders. Rees draws on interviews collected over the years for his TV programmes, often previously unpublished. Like a documentary, the book frequently cuts from the narrator to eyewitnesses, adding immediacy and poignancy (though one would like to learn more about some of the individuals).

We meet ordinary Germans who fell in line with a regime that promised them peace and prosperity

We meet ordinary Germans who fell in line with a regime that promised them peace and prosperity. Interviewed decades after the destruction of the Third Reich, some still looked back wistfully to the days before the war. “You saw the unemployed disappearing from the streets,” recalled Erna Krantz from Bavaria. “There was order and discipline … It was, I thought, a better time”.

We meet callous perpetrators such as Wolfgang Horn, a soldier who did not think twice about burning down a Russian village, because the locals were “too primitive for us”, and foreign collaborators such as Michal Kabáč, a Slovak guard who stole from Jews and forced them on to deportation trains: “We had a good salary, accommodation and canteen. We could not complain.”

And we meet Jews from across Europe, each with their own heartbreaking story to tell. Take Eva Votavová, who was deported to Auschwitz with her family in summer 1942. She last saw her father, looking “sad and hopeless”, after the SS separated men and women on arrival. Sometime later, she found a pair of glasses amid a pile of remains, and began to cry. “I knew they were my mother’s.” Eva herself fell ill with typhoid fever, and lost the glasses she had hidden. “That is how I lost the last memory of my mother.” Rees helps to recover this memory, and the memories of other victims, survivors of “a crime of singular horror in the history of the human race”.

• Nikolaus Wachsmann’s KL: A History of the Nazi Concentration Camps is published in paperback by Abacus. The Holocaust: A New History is published by Viking. To order a copy for £21.25 (RRP £25) go to bookshop.theguardian.com or call 0330 333 6846. Free UK p&p over £10, online orders only. Phone orders min p&p of £1.99.