Let F be a graph. We say that a hypergraph H contains an {induced Berge} F if there exists an injective mapping f from the edges of F to the hyperedges of H such that if xy \in E(G), then f(xy) \cap V(F) = {x,y}. We show that the maximum number of edges in an $r$-uniform hypergraph with no induced Berge F is strongly related to the generalized Turán function ex(n,K_r, F). (I.e., the maximum number of K_r's in an F-free graph on n vertices).