The development of Best Linear Unbiased Prediction (BLUP) by Henderson (1973) equipped animal breeders with a tool to evaluate animals for their genetic potential much more efficiently than had been previously possible. However, BLUP requires solving a large system of simultaneous equations (the Mixed Model Equations: MME) which is computationally demanding. It also requires knowledge of the covariances among all random effects. Covariances among additive genetic effects are described by a function of the Numerator Relationship Matrix (A). A simple method to compute its inverse (A⁻¹) was necessary before BLUP could be adopted widely. Such a method was found by Henderson (1975, 1976) whereby A⁻¹ could be written directly from a list of animals, their pedigrees and the inbreeding coefficients of all parents. Now, BLUP is the method of choice for genetic evaluation of livestock and its use is widespread across species and countries. This thesis presents some computer algorithms which reduce the time and memory required to evaluate livestock using BLUP. In six chapters, questions relating to the building of the numerator relationship matrix and its inverse, computing inbreeding coefficients and storing and solving the MME are addressed.