  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 Bra{\ss}-K\'{a}rolyi-Valtr, Capoyleas-Pach and
Aronov-Dujmovi\v{c}-Morin-Ooms-da Silveira. We also provide a new
generalization of the Erd\H os-Ko-Rado theorem in the ordered setting.
