  Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if
there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$
for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of
hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no
subhypergraph isomorphic to any Berge-$F$ is denoted
$\ex_r(n,\textrm{Berge-}F)$. In this paper we establish new upper and lower
bounds on $\ex_r(n,\textrm{Berge-}F)$ for general graphs $F$, and investigate
connections between $\ex_r(n,\textrm{Berge-}F)$ and other recently studied
extremal functions for graphs and hypergraphs. One case of specific interest
will be when $F = K_{s,t}$. Additionally, we prove a counting result for
$r$-graphs of girth five that complements the asymptotic formula $\textup{ex}_3
(n , \textrm{Berge-}\{ C_2 , C_3 , C_4 \} ) = \frac{1}{6} n^{3/2} + o( n^{3/2}
)$ of Lazebnik and Verstra\"{e}te [{\em Electron.\ J. of Combin}. {\bf 10},
(2003)].
