This paper documents the use of a massively parallel computer, specifically the Connection Machine CM-5, to solve the Boltzmann equation to model one-dimensional shock wave structure, a boundary layer, and general 3-D flow fields. The Bhatnagar-Gross-Krook (BGK) model for the collision term combined with a finite difference scheme was used to model the flow. This collision terms requires accurate knoweldge of the density, temperature, and mean velocity. Great care must be taken in their calculation to insure conservation, which proved to be the most difficult part. The algorithm, however, is well suited to the Connection Machine, and accurate results were obtained with great efficiency. The 1-D version of the code (which actually models a 5-D problem in phase space) was optimized for the CM-5 and sustained 61 gigaflops on a 1024-node CM-5.