Percolation Scheduling (PS) is a new technique for compiling programs into parallel code. It attempts to overcome problems that limit the effectiveness and applicability of currently available techniques. PS globally rearranges code past basic block boundaries. Its core is a small set of simple, primitive program transformations defined in terms of adjacent nodes in a program graph. These transformations constitute the lowest level in a system of transformations and guidance rules. Higher levels of this hierarchy control and enhance the applicability of the core transformations and enable us to exploit both fine grained and coarse parallelism. Unlike other, more ad hoc approaches, PS is based on rigorous definitions of the computational model and of the core transformations. The correctness and termination of the transformations is proven here. The completeness of the transformations is also discussed. As a result our system implementation can proceed on a sound basis. In particular, PS enjoys greater adaptability and independence between the levels than would be possible otherwise. This has greatly facilitated the implementation of the first prototype version of our compiler, which is now complete. This paper describes PS in detail. The correctness aspects as well as illustrations of the effectiveness of our techniques are presented. Architectures which may benefit from the use of PS are also discussed.