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 determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braß-Károlyi-Valtr, Capoyleas-Pach and Aronov-Dujmovič-Morin-Ooms-da Silveira. We also provide a new generalization of the Erd\H os-Ko-Rado theorem in the ordered setting.