Generalized inverses of matrices are of great importance in the resolution of linear systems and have been extensively studied by many researchers. A collection of some results on generalized inverses of matrices over commutative rings has been provided by K. P. S. Bhaskara Rao (2002). In this thesis, we consider constructing algorithms for finding generalized inverses and generalizing the results collected in Rao's book to the non-commutative case. We first construct an algorithm by using the greatest common divisor to find a generalized inverse of a given matrix over a commutative Euclidean domain. We then build an algorithm for finding a generalized inverse of a matrix over a non-commutative Euclidean domain by using the one-sided greatest common divisor and the least common left multiple. Finally, we explore properties of various generalized inverses including the Moore-Penrose inverse, the group inverse and the Drazin inverse in the non-commutative case.