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 Turán-type problems, focusing on recent developments.