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Summary Massively parallel single-instruction, multiple-data (SIMD) computers have shown much promise in solving numerically intensive problems ranging from molecular modeling to computational fluid dynamics. Much of the research on the use of parallel computing for reservoir simulation, however, has been limited to coarse-grain, shared-memory computers or medium-grain hypercubes. We report here on research performed on a massively parallel SIMD computer with 65, 536 processors. This work addresses issues in machine architecture, programming environment, and formulation of a reservoir simulator on SIMD computers. Toward this end, a three-phase, 3D, implicit-pressure, explicit-saturation (IMPES) compositional simulator was developed in FORTRAN 90. Problems associated with reservoir simulation on SIMD computers (such as an appropriate data structure for the SIMD architecture, treatment of multiple fluid phases, data communication, and the matrix solution) are addressed in this paper. In addition, table lookup poses a unique problem for SIMD computers because storage of large tables on each processor is impractical, as is the use of the front-end computer to perform this function. The problem is overcome through a novel use of local memory access. We found that massively parallel computers can be used to run very large reservoir models and that the cost of data transfer between processors need not be prohibitive. Reservoir models with two hydrocarbon components and up to 2, 097, 152 gridblocks were run successfully with this simulator. Computational speeds on the order of 1 GFLOP's (1 giga floating point operations) were achieved for the generation of the Jacobian matrix and the matrix solution with 65, 536 processors.