We present efficient and practical algorithms for a large, distributed system of processors to achieve reliable computations in a secure manner. Specifically, we address the problem of computing a general function of several private inputs distributed among the processors of a network, while ensuring the correctness of the results and the privacy of the inputs, despite accidental or malicious faults in the system. [...] Our algorithms maintain a low cost in local processing time, are the first to achieve optimal levels of fault-tolerance, and most importantly, have low communication complexity. In contrast to the best known previous methods, which require large numbers of rounds even for fairly simple computations, we devise protocols that use small messages and a constant number of rounds regardless of the complexity of the function to be computed. Through direct algebraic approaches, we separate the communication complexity of secure computing from the computational complexity of the function to be computed. We examine security under both the modern approach of computational complexity-based cryptography and the classical approach of unconditional, information-theoretic security. We [...] support formal proofs of claims to security, addressing an important deficiency in the literature. Our protocols are provably secure. In the realm of information-theoretic security, we characterize those functions which two parties can compute jointly with absolute privacy. We also characterize those functions which a weak processor can compute using the aid of powerful processors without having to reveal the instances of the problem it would like to solve. Our methods include a promising new technique called a locally random reduction, which has given rise not only to efficient solutions for many of the problems considered in this work but to several powerful new results in complexity theory.