The recent trend toward increased parallelism in supercomputer architectures, together with the need to use existing supercomputer resources more efficiently, has motivated this study of parallel algorithms designed specifically for implementation on parallel machines. Users of parallel machines like the ICL Distributed Array Processor (DAP) are all too familiar with the problem of having to reformulate their algorithms to incorporate the inherent parallelism of the DAP architecture. The individual processing elements of the DAP are comparatively slow, and the full potential of such a machine is only realized if the algorithm used is highly parallel. The very fast processing capabilities of pipeline machines like the Cray-1 or the Cyber 205 tend to obscure this point somewhat, and in fact most users of these machines are content with using existing codes, vectorized where possible. Such a strategy, however, is not appropriate for the parallel processors like the ICL DAP, AMT DAP, the Goodyear MPP, GEC GRID, etc. The application of parallel algorithms to the solution of fluid dynamics problems is considered. Most of the work concerns the solution of the two- and three-dimensional incompressible Navier-Stokes equations, steady and unsteady. The results are relevant for the above-mentioned parallel machines, but the methods can be adapted for use on vector machines. Concerning parallel algorithms, the easiest and possibly those with the widest potential application are relaxation techniques based on red-black (R-B) ordering. The use of R-B successive overrelaxation, R-B LSOR, etc., on two- and three-dimensional cavity flows, unsteady channel, and boundary layer flows at high Reynolds numbers is considered. While these methods are extremely problem dependent, in the favorable cases they can be as competitive as more sophisticated serial “vectorized” algorithms. As far as parallel algorithms are concerned, the most important criteria are what is the cost per iteration, and how fast do they converge. For application on serial machines, the latter factor seems to dominate.