  Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex
set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a
bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we
have $e \subset f(e)$. This generalizes the established definitions of "Berge
path" and "Berge cycle" to general graphs. For a fixed graph $G$ we examine the
maximum possible size (i.e.\ the sum of the cardinality of each edge) of a
hypergraph with no Berge-$G$ as a subhypergraph. In the present paper we prove
general bounds for this maximum when $G$ is an arbitrary graph. We also
consider the specific case when $G$ is a complete bipartite graph and prove an
analogue of the K\H ov\'ari-S\'os-Tur\'an theorem.
