An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and discover a general partitioning phenomenon which allows us to determine the order of magnitude of the extremal function for various ordered and convex geometric hypergraphs. A special case is the ordered $n$-vertex $r$-graph $F$ consisting of two disjoint sets $e$ and $f$ whose vertices alternate in the ordering. We show that for all $n \geq 2r + 1$, the maximum number of edges in an ordered $n$-vertex $r$-graph not containing $F$ is exactly \[ {n \choose r} - {n - r \choose r}.\] This could be considered as an ordered version of the Erdős-Ko-Rado Theorem, and generalizes earlier results of Capoyleas and Pach and Aronov-Dujmovič-Morin-Ooms-da Silveira.