Erdos and Szemeredi conjectured that if A is a set of n positive integers, then either the sumset A+A or the product set A.A will contain at least n^{2-epsilon} distinct numbers. This problem is still unsolved, but important progress has been made recently by Bourgain, Chang, Katz, Konyagin, Solymosi, and Tao using methods from both combinatorial number theory and harmonic analysis. They obtain results on sum and product sets of complex numbers as will as on sum and product sets in finite fields. This will be an expository talk on their work.