  Let $F = (U,E)$ be a graph and $\mathcal{H} = (V,\mathcal{E})$ be a
hypergraph. We say that $\mathcal{H}$ contains a Berge-$F$ if there exist
injections $\psi:U\to V$ and $\varphi:E\to \mathcal{E}$ such that for every
$e=\{u,v\}\in E$, $\{\psi(u),\psi(v)\}\subset\varphi(e)$. Let $ex_r(n,F)$
denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$
vertices which does not contain a Berge-$F$.
  For small enough $r$ and non-bipartite $F$, $ex_r(n,F)=\Omega(n^2)$; we show
that for sufficiently large $r$, $ex_r(n,F)=o(n^2)$. Let $thres(F) = \min\{r_0
:ex_r(n,F) = o(n^2) \text{ for all } r \ge r_0 \}$. We show lower and upper
bounds for $thres(F)$, the uniformity threshold of $F$. In particular, we
obtain that $thres(\triangle) = 5$, improving a result of Gy\H{o}ri.
  We also study the analogous problem for linear hypergraphs. Let $ex^L_r(n,F)$
denote the maximum number of hyperedges in an $r$-uniform linear hypergraph on
$n$ vertices which does not contain a Berge-$F$, and let the linear unformity
threshold $thres^L(F) = \min\{r_0 :ex^L_r(n,F) = o(n^2) \text{ for all } r \ge
r_0 \}$. We show that $thres^L(F)$ is equal to the chromatic number of $F$.
