High Performance Fortran (HPF) is a set of extensions to Fortran expressing parallel execution at a relatively high level. For the thousands of scientists, engineers, and others who wish to take advantage of the power of both vector and parallel supercomputers, five of the principal authors of HPF have teamed up here to write a tutorial for the language. There is an increasing need for a common parallel Fortran that can serve as a programming interface with the new parallel machines that are appearing on the market. While HPF does not solve all the problems of parallel programming, it does provide a portable, high-level expression for data- parallel algorithms that brings the convenience of sequential Fortran a step closer to today''s complex parallel machines.