For G=(V,E) a graph on n=|V| vertices and random sets L(v) (for each v in V), what conditions suffice to make G L-colorable with high probability as n goes to infinity? In joint work with Jeff Kahn, in Chapters 1-9, the following conditions are shown to be sufficient. For any d>0, with D equal to the maximum degree of G and S(v) given sets of size D+1 (for each v in V), with L(v) drawn uniformly at random from the (1+d)ln(n)-subsets of S(v) (for each v in V, independently of other choices), the probability that G is L-colorable converges to 1 as n goes to infinity.