This text presents recent advances in algorithmic ways of introducing fault-tolerance in multiprocessors under the constraint of preserving efficiency. The difficulty associated with combining fault-tolerance and efficiency is that the two have conflicting means: fault-tolerance is achieved by introducing redundancy, while efficiency is achieved by removing redundancy. This monograph demonstrates how in certain models of parallel computation it is possible to combine efficiency and fault-tolerance. It also shows how it is possible to develop efficient algorithms without concern for fault-tolerance, and then correctly and efficiently execute these algorithms on parallel machines whose processors are subject to arbitrary dynamic fail-stop errors. The efficient algorithmic approaches to multiprocessor fault-tolerance presented in this monograph make a contribution towards bridging the gap between the abstract models of parallel computation and realizable parallel architectures. The monograph synthesizes work that was presented in recent symposia and published in refereed journals by the authors and other leading researchers.