A well-known problem in the design of operating systems is the selection of a resource allocation policy that will prevent deadlock. Deadlock is the situation in which resources have been allocated to various tasks in such a way that none of the tasks can continue. The various published solutions have been somewhat restrictive: either they do not handle the problem in sufficient generality or they suggest policies which will on occasion refuse a request which could have been safely granted. Algorithms are presented which examine a request in the light of the current allocation of resources and determi.~ e whether or not the granting of the request will introduce the possibility of a deadlock. Proofs given in the appendixes show that the conditions imposed by the algorithms are both necessary and sufficient to prevent deadlock. The algorithms have been successfully used in the THE system.