In this chapter, we deal with extremal problems for uniform hypergraphs that are motivated by Turán’s graph theorem. These extremal problems are substantially more difficult than for graphs, and indeed the extremal problem for hypergraph cliques remains unsolved, even asymptotically as the number of vertices grows. Perhaps the most notorious conjecture is the Turán conjecture from 1941 about the maximum density of a triple system on n points that contains no copy of the tetrahedron. This chapter explores a variety of Turán-type problems for hypergraphs while introducing the important methods in this area.