The goal of this survey is to give a view of the state-of-the-art of computing the Singular Value Decomposition (SVD) of dense and sparse matrices, with some emphasis on those schemes that are suitable for parallel computing platforms. For dense matrices, we survey those schemes that yield the complete decomposition, whereas for sparse matrices we survey schemes that yield only the extremal singular triplets. Special attention is devoted to the computation of the smallest singular values which are normally the most difficult to evaluate but which provide a measure of the distance to singularity of the matrix under consideration. Also, a parallel method for computing pseudospectra, which depends on computing the smallest singular values, is presented at the conclusion of the survey.