Introduction This research is concerned with the development of very fast parallel algorithms, ones faster than those available through normal programming techniques or standard parallel computers. Algorithms have been developed for problems in geometry, graph theory, arithmetic, sorting, and image processing. The computing models that these algorithms have been developed for are concurrent read concurrent write parallel random access machines (CRCW PRAMs ), and reconfigurable meshes (rmeshes, defined below). For CRCW PRAMS, our work has shown that by combining randomization with the use of some extra memory, one can solve some problems far faster than they can be solved if only randomization is used. We have developed ultrafast algorithms for several problems, where by ultrafast algorithm we mean a parallel algorithm with an input of size n which uses at most a linear number of processors and finishes in poly-log