Let n be a large prime. A set A of residues modulo n is zero-sum-free if no subsetsum of A is divisible by n. Zero-sum-free sets have been studied for a long time but little was know about the following fundamental question: How many zero-sum-free sets are there ?In this talk, we shall present a sharp answer to this question, using new results about long arithmetic progressions in sumsets. In fact, we are able to characterize zero-sum-free sets: the main (and natural) reason for a set to be zero-sum-free is that the sum of its elements is less than n. (joint work with E. Szemeredi)