Monte Carlo particle transport methods are extensively used, due to the generality and accuracy of these methods. In Monte Carlo methods hundreds of thousands or millions of particle histories are simulated using random numbers, highly accurate representations of particle reaction probabilities and exact models for 3-D geometries. The principal limitations for Monte Carlo methods are the requirement to simulate many particles to achieve an acceptable statistical uncertainty. This requirement provides ample incentive to utilize the computational power of modern vector and parallel supercomputers. Monte Carlo particle transport algorithms are inherently parallel because each particle history can be simulated independently and concurrently on separate processors. Monte Carlo particle transport codes have been successfully implemented on a number of different computational platforms [Mar91]. For adaptation to vector computers, the computational algorithm should be changed from a history-based scheme to an event-based scheme [Bro81, Bro85, Bro86, Cha85, Mar86, Mar87a]. Even though this requires restructuring of all data and extensive recoding, successful vectorized codes have provided gains of 10 - 100X or greater in computational speed. Moreover, once vectorized, it is relatively easy to parallelize a Monte Carlo code across multiple vector processors [Bob84]. Many research organizations have resisted the vectorization of production Monte Carlo codes, due to the considerable investments in time and manpower involved. On the other hand, the relative ease of parallelization and continuing decrease in costs of parallel computers, both massively parallel processors (MPPs) and distributed workstations, makes it attractive to adapt Monte Carlo codes to parallel computers [Mar87b, Mar93]. Monte Carlo algorithms are ideally suited for distributed and shared memory MIMD (Multiple Instruction, Multiple Data) parallel processors, because of the inherent parallelism involved in the fundamental algorithm. If enough memory is available to each processor, the Monte Carlo code and data can be replicated, and each processor can independently follow a portion of the total particles. This is possible due to the statistical independence of the particle histories. Monte Carlo particle transport codes have been demonstrated to be efficient and effective on both MPPs and distributed workstations. Two important issues involving parallel random number generators and reproducibility of results on parallel computers will be discussed in a later chapter of this thesis. We will also discuss our effort to implement various Monte Carlo algorithms on the two parallel computers (KSR-1 and IBM-SP2) operated by the Center for Parallel Computing at the University of Michigan and on the BBN TC2000 at the Lawrence Livermore National Laboratory.