This paper is divided into three parts: an introduction to the Hybrid Monte Carlo (HMC) algorithm; a detailed discussion of its implementation for Quantum Chromodynamics (QCD) in CMIS on the Connection Machine (CM-2); and a summary of recent analytic results on the behavior of HMC for free field theory. We present a detailed description of the architecture of the CM-2 and explain how it can execute the Conjugate Gradient (CG) solver for HMC at 6.35 GFlops/s.