  The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained
from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$
disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets.
Let $ex_r(n,F)$ denote the maximum number of edges in an $r$-uniform hypergraph
with $n$ vertices not containing any copy of the $r$-uniform hypergraph $F$.
Many problems in extremal set theory ask for the determination of $ex_r(n,G^+)$
for various graphs $G$. We survey these Tur\'an-type problems, focusing on
recent developments.
