How did the infamous Josef Mengele escape punishment?

The atrocities of the concentration camp at Auschwitz–Birkenau are now universally known, but it is still almost beyond belief that Auschwitz could exist in modern Europe. The history of the camp is a comparatively recent one: construction began in April 1940, less than 80 years ago, and the first victims died there, or were killed, not long after. Some of the people who were transported to the camp managed somehow to survive the war and a few of them are still alive today. Perhaps we shy away from knowing too much about what they suffered because the totality of awfulness is beyond description and the world is still contemplating that.

There is one particular matter of which we have all heard, but on which we choose to look away for fear of what we shall be told, and that is the exact nature of the career of Josef Mengele, and how he escaped.