We know since almost a century that a ball can be decomposed into five pieces and these pieces rearranged so as to produce two balls of the same size as the original.

This apparent paradox has led von Neumann to the notion of amenability which is now much studied in many areas of mathematics.

However, the initial paradox has remained tied down to an elementary property of free groups of rotations for most of the 20th century. I will describe recent progress leading to new paradoxical groups.