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In the past several years, many new parallel and multiple processor architectures have been developed, all aimed at increasing machine performance by an order of magnitude or more. Several classification schemes for various parallel and multiprocessing architectures are examined. An overview of current parallel and multiprocessor-based computers is given. Flynn's classification scheme is based on 2 streams: 1. the instruction stream defined as a sequence of instructions performed by a computer, and 2. a data stream defined as a sequence of data used to execute an instruction stream. Single instruction streams, multiple data streams and multiple instruction multiple data (MIMD) machines belong to the class of parallel and multiprocessor architectures. MIMD machines are divided into shared memory MIMD, message passing MIMD, and hybrid machines.