Increasing use will be made of multifault-(n-fault-) tolerant systems, i.e. systems whose performance is unaffected by any arbitrary combination ofnfaults. The paper presents a family of fundamental fault-tolerant systems using standby spares. Recursive algebraic expressions are given both for the smallest combinations of disabling failures and for the probabilistic reliabilities of each system in the family. It is shown that multifault tolerance by itself is not exactly synonomous with a best reliability or a best cost/benefit parameter. The ‘crossover’ surface of any redundancy is defined as the infinite set of all combinations of component reliabilities (as they decrease) for which the reliability of the redundant system has deteriorated to the same reliability as the nonredundant system. For component reliabilities outside the ‘crossover’ surface, the presence of the redundancy further decreases system reliability. This fact is established numerically forn=6. The 3-failure-tolerant system was proposed for an aerospace application, but in a slightly modified form. This modified form would tolerate only two failures of the most highly reliable components, but three failures of the least reliable components.