Consider a function f defined on the set of graphs on a fixed set of n vertices. Assume that f satisfies the following “continuity” condition: (^*) |f(G) - f(G′)| ≦ 1 whenever G and G′ differ by at most one edge. (An example of such a function f is the maximum matching of the graph G) Then we prove that in either of the classical models and G _n,p of random graphsf is very concentrated around its expectation. (The concentration bounds we obtain are of optimal order.).