When the Second World War ended, the people of liberated Europe celebrated their freedom from Nazi tyranny. Their years of suffering had ended, but for millions of Germans, the end of the conflict opened a new and terrible chapter. They suffered the appalling violence meted out to the defeated, especially to those ethnic Germans who had lived peacefully for centuries in neighbouring countries". Savage Peace uses rare and unseen archive film, the documentary tells a harrowing story of vengeance against German civilians, which mirrored some of the worst cruelty of the Nazi occupiers during the years of war. The Savage Peace includes the unique testimony of eyewitnesses and victims, who recall the horrors with searing clarity, their memories undimmed 70 years after the events took place. This a story that has, until now, been untold amidst the justified celebration of an end to an unspeakable tyranny. But as the writer George Orwell said, the treatment of the defeated Germans was a terrible crime that has gone unpunished.

When ancient Greece fell into decline, mathematical progress stagnated as Europe entered the Dark Ages, but in the East mathematics reached new heights. In the second episode, Du Sautoy explores how maths helped build imperial China and discovers how the symbol for the number zero was invented in India. He also looks at the Middle Eastern invention of algebra and how mathematicians such as Fibonacci spread Eastern knowledge to the West.

The Red Sea is one of the youngest oceans in the world and one of the warmest. The northern region, with its diverse array of marine life, is one of the most visited dive locations on earth while the southern end remains virtually unexplored. It's a global hotspot for marine biology and an important trade route throughout human history, linking the trade goods of India and the Far East with the markets of Egypt and Europe.

Dr Alice Roberts looks at our ancestors' seemingly impossible journey to Australia. Miraculously preserved footprints and very old human fossils buried in the outback suggest a mystery: that humans reached Australia almost before anywhere else. How could they have travelled so far from Africa, crossing the open sea on the way, and do it thousands of years before they made it to Europe? The evidence trail is faint and difficult to pick up, but Alice takes on the challenge. In India, new discoveries among the debris of a super volcano hint that our species started the journey much earlier than previously thought, while in Malaysia, genetics points to an ancient trail still detectable in the DNA of tribes today. Alice travels deep into the Asian rainforests in search of the first cavemen of Borneo and tests out a Stone Age raft to see whether sea travel would have been possible thousands of years ago, before coming to a powerful conclusion.

Professor Robert Bartlett explores the impact of the Normans on southern Europe and the Middle East. The Normans spread south in the 11th century, winning control of southern Italy and the island of Sicily. There they created their most prosperous kingdom, where Christianity and Islam co-existed in relative harmony and mutual tolerance. It became a great centre of medieval culture and learning. But events in the Middle East provoked the more aggressive side of the Norman character. In 1095, the Normans enthusiastically answered the Pope's call for holy war against Islam and joined the first crusade. They lay siege to Jerusalem and eventually helped win back the holy city from the muslims. This bloody conquest left a deep rift between Christianity and Islam which is still being felt to this day.

Nearly half a million years ago, the most advanced human yet roams Europe. Strong and powerful, Homo heidelbergensis are fierce hunters, use sophisticated tools and live in close-knit family groups. Over 200,000 years they become split into two populations by extremes of weather and environment and evolve separately into two very different species. In the North are the Neanderthals, whose physical power and resilience is the key to surviving in ice age Northern Europe. About 140,000 years ago, Africa is in the grip of a devastating drought, and something remarkable has happened to the descendants of heidelbergensis who live there. The combination of environment and chance has bred in them a unique ability that will change the course of human history. It will be this small band of southern survivors, perhaps numbering just a few tens of thousands, who will come to dominate the world and be known as Homo sapiens.