In this paper, we describe efficient MapReduce simulations of parallel algorithms specified in the BSP and PRAM models. We also provide some applications of these simulation results to problems in parallel computational geometry for the MapReduce framework, which result in efficient MapReduce algorithms for sorting, 1-dimensional all nearest-neighbors, 2-dimensional convex hulls, 3-dimensional convex hulls, and fixed-dimensional linear programming. For the case when reducers can have a buffer size of $B=O(n^ε)$, for a small constant $ε>0$, all of our MapReduce algorithms for these applications run in a constant number of rounds and have a linear-sized message complexity, with high probability, while guaranteeing with high probability that all reducer lists are of size $O(B)$.