  For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number
of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of
$F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central
problem in extremal combinatorics. While for graphs ($k=2$) this problem is
well understood, due to the work of Mantel, Tur\'an, Erd\H{o}s, Stone,
Simonovits and many others, only very little is known for $k$-uniform
hypergraphs for $k>2$. Already the case when $F$ is a $k$-uniform hypergraph
with three edges on $k+1$ vertices is still wide open even for $k=3$.
  We consider variants of such problems where the large hypergraph $H$ enjoys
additional hereditary density conditions. Questions of this type were suggested
by Erd\H{o}s and S\'os about 30 years ago. In recent work with R\"odl and
Schacht it turned out that the regularity method for hypergraphs, established
by Gowers and by R\"odl et al. about a decade ago, is a suitable tool for
extremal problems of this type and we shall discuss some of those recent
results and some interesting open problems in this area.
