We address the task of doing computational physics problems on a computer which has a highly parallel architecture. Our experience has been gained on the ICL DAP, the first commercially available parallel processor with over a thousand processing elements, but the results have relevance to all computers of this generic class. The DAP itself is described, along with some aspects of the extension to the Fortran language that have been made in order for the user to exploit the DAP's parallelism (or concurrency). Again although herein specific, these language constructs are necessary in one form or another on any such highly parallel computer. The physics we cover starts with molecular dynamics as the concepts here are intuitively obvious--given a governing function it is possible to follow the time development of a system initially randomized, this motion being then a simulation of the motion in a real-life system. The first system models the approach to melting of a molecular solid, naphthalene. It is shown that molecules enjoy increasing reorientational freedom in a narrow temperature interval, and show behavior as suggested from experimental work. The second system modeled is SF6, which undergoes a phase transition from plastic to partially plastic crystalline, and then another transition to a truly crystalline phase. The simulation allows us to watch the organizational behavior of the molecules while the system is in transition, giving a very realistic model at the atomic level of solid-state phase transitions. We continue with lattice gauge theory (or quantum chromodynamics) which governs the behavior of quarks and gluons. The method used here is Monte Carlo, a method where statistical configurations are chosen by the use of pseudo-random numbers from the appropriate distribution. A lattice is overlaid on a region of space-time, and the function represented on this lattice leads to many results which test the basic theory, such as the mass spectrum for baryons and mesons. This is an area of physics of fundamental importance, and may yet need an increase of computational resource of about a factor of 100 before its full potential is realized. For this reason we treat the problem in some detail. In conclusion we look at the three-dimensional Ising model and its study through real-space renormalization. Because this is a very simply defined problem it is ideal for gaining insight into the methods of using the renormalization group, methods which will have to be fully understood before the QCD results can be interpreted in full.