How is algorithmic randomness related to the classical theory of uniform distribution? In this talk we consider the definition of Martin-Löf  randomness for real numbers in terms of uniform distribution of sequences.  We present a necessary condition for a real number to be Martin-Löf random, and a strengthening of that condition which is  sufficient for Martin-Löf randomness. For this strengthening we define a notion of uniform distribution relative to the computably enumerable open subsets of the unit interval. We call the notion Sigma^0_1-uniform distribution.

This is joint work with Serge Grigorieff and Theodore Slaman.