Abstract: Let A be a finite subset of real numbers. If A is an arithmetic progression, then |A+A|=2|A|-1, and |A*A| > |A|^{2-\epsilon}, where A+A={a+a': a,a' \in A} is the sum-set of A and A*A={aa' : a,a' \in A} is the product-set. ( |S| denotes the cardinality of S.)

Similarly, if B is an arithmetic progression and A={2^n: n \in B}, then the product-set is small, |A*A|=2|A|-1, but the sum-set is large, |A+A| > c|A|^2.

Erdos and Szemeredi conjectured that the sum-set or the product-set should be always large, max (|A+A|,|A*A|) > c|A|^{2-\delta}. ( \delta goes to 0 as |A| goes to infinity.) In this talk we will summarize the recent results and further research. directions.