Consideration is given to the two main methods of numerical analysis which are currently used for solving problems in both small and large displacement elasticity, and in elasto-plastic problems. These are the finite element method, and the dynamic relaxation method. The former is of greater applicability than the latter, and is therefore treated in greater detail, so as to include methods for problems in solid mechanics which involve either one-, two-, or three-dimensional continua, as well as some problems in fluid mechanics such as flow through porous media. Recent work in this field tends in the direction of more efficient utilization of the methods which have been developed during the past ten years. This means greater attention to computer storage organization and to more efficient solution routines. Essentially the choice here is between direct elimination and iteration methods, each of these two main categories having many varieties. The available methods are briefly discussed and some indication given of the more efficient computer methods.