The Turan function $ex(n,F)$ of a k-graph $F$ is the maximum number of edges in an $F$-free k-graph on $n$ vertices. This problem goes back to the fundamental paper of Turan from 1941 that solved it for complete graphs (k=2). Unfortunately, very few non-trivial instances of the problem have been solved when we consider hypergraphs (k>2).We survey some recent results and methods on the hypergraph Turan function. In particular, we discuss the so-called stability approach that greatly helps in obtaining exact results from asymptotic computations (for example, those that use flag algebras or graph limits).