Three fundamental problems of mechanical reliability are addressed. (1) computing the probability of failure, p(f), of a component having design factors with known statistical distributions and a limit state with a closed form algebraic expression (2) computing the probability of failure of a component having design factors with known distributions and a limit state which can only be expressed by a computer algorithm, and (3) deriving safety check expressions in a "design by reliability" approach. An algorithm for generating estimates of p(f) is presented. The method is an extension of, and demonstrated to be a significant improvement to, the widely used Rackwitz-Fiessler (R-F) method--a fast and efficient numerical method for performing reliability analysis. Comparisons were made for numerous examples, it was found that the error in p(f), using the proposed method, is typically about half of the error in R-F estimates. A method was proposed for computing p(f) when the relationship between design factors can be defined only using a computer algorithm, e.g., finite element analysis. A second order polynomial is constructed, using a simple curve fitting routine, to approximate the limit state in the neighborhood of the design point (i.e., a point close to the most likely value of the design variables at failure). Then the R-F method can be applied easily. It is demonstrated that this scheme is much faster than the Monte Carlo method in producing reasonable estimates of p(f). Methods of deriving safety check expressions for design codes and design criteria documents are studied. A Level I format employing partial safety factors derived from Level II methods is used to construct the safety check expressions which are suitable for code development. The procedures are demonstrated using numerous examples which include the problems where the limit states are complicated, i.e., the limit states are not explicitly defined.