The Turán function ex (n, F) of a graph F is the maximum number of edges in an F-free graph with n vertices. The classical results of Turán and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine h F (n, q), the minimum number of copies of F that a graph with n vertices and ex (n, F)+ q edges can have. We determine h F (n, q) asymptotically when F is color-critical (that is, F contains an edge whose deletion reduces its chromatic number) and q= o (n 2). Determining the exact value of h F (n, q) seems rather difficult. For example, let c 1 be the limit superior of q/n for which the extremal structures are obtained by adding some q edges to a maximum F-free graph. The problem of determining c 1 for cliques was a well-known question of Erdős that was solved only decades later by Lovász and Simonovits. Here we prove that c 1> 0 for every color-critical F. Our approach also allows us to determine c 1 for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge.