We introduce the notion of a geometric hypergraph. We discuss the extremal result in the geometric hypergraph, i.e., how many hyperedges a uniform geometric hypergraph can have if it does not contain any crossing pair of hyperedges? We then discuss a lower bound on the number of crossing pairs of hyperedges in a complete geometric d-hypergraph. We also discuss the maximum crossing number of a complete geometric d-hypergraph for d<=4. We further discuss open questions in this regard and their connection to the polytope theory.