Two important issues in systolic array designs are addressed: How is fault tolerance provided in systolic arrays to enhance the yield of wafer-scale integration implementations? And, how are efficient systolic arrays with two levels of pipelining designed? (The first level refers to the pipelined organization of the array at the cellular level, and the second refers to the pipelined functional units inside the cells.) The fault-tolerant scheme proposed replaces defective cells with clocked delays. This has the distinct characteristic that data can flow through the array with faulty cells at the original clock speed. It is shown that both the defective cells under this fault-tolerant scheme and the second-level pipeline stages can simply be modeled as additional delays in the data paths of “generic” systolic designs. The mathematical notion of a cut is introduced to solve the problem of how to allow for these extra delays while preserving the correctness of the original systolic array designs. The results obtained by applying these techniques are encouraging. When applied to systolic arrays without feedback cycles, the arrays can tolerate large numbers of failures (with the addition of very little hardware) while maintaining the original throughput. Furthermore, all of the pipeline stages in the cells can be kept fully utilized through the addition of a small number of delay registers. However, adding delays to systolic arrays with cycles typically induces a significant decrease in throughput. In response to this, a new class of systolic algorithms has been derived in which the data cycle around a ring of processing cells. The systolic ring architecture has the property that its performance degrades gracefully as cells fail. Use of the cut theory and ring architectures for arrays with feedback gives effective fault-tolerant and two-level pipelining schemes for most systolic arrays. As a side effect of developing the ring architecture approach, several new systolic algorithms have been derived. These algorithms generally require only one-third to one-half of the number of cells used in previous designs to achieve the same throughput. The new systolic algorithms include ones for LU-decomposition, QR-decomposition, and the solution of triangular linear systems.