  Numerous problems in extremal hypergraph theory ask to determine the maximal
size of a $k$-uniform hypergraph on $n$ vertices that does not contain an
`enlarged' copy $H^+$ of a fixed hypergraph $H$. These include well-known
problems such as the Erd\H{o}s-S\'{o}s `forbidding one intersection' problem
and the Frankl-F\"{u}redi `special simplex' problem.
  We present a general approach to such problems, using a `junta approximation
method' that originates from analysis of Boolean functions. We prove that any
$H^+$-free hypergraph is essentially contained in a `junta' -- a hypergraph
determined by a small number of vertices -- that is also $H^+$-free, which
effectively reduces the extremal problem to an easier problem on juntas. Using
this approach, we obtain, for all $C<k<n/C$, a complete solution of the
extremal problem for a large class of $H$'s, which includes the aforementioned
problems, and solves them for a large new set of parameters.
  We apply our method also to the 1974 Erd\H{o}s-Chv\'{a}tal simplex
conjecture, which asserts that for any $d < k \leq \frac{d}{d+1}n$, the maximal
size of a $k$-uniform family that does not contain a $d$-simplex (i.e., $d+1$
sets with empty intersection such that any $d$ of them intersect) is
${{n-1}\choose{k-1}}$. We prove the conjecture for all $d$ and $k$, provided
$n>n_0(d)$.
