The ordinary Singular Value Decomposition (SVD) is widely used in statistical and signal processing computation, both for the insight it provides into the structure of a linear operator, and as a technique for reducing the computational word length required for least-squares solutions and certain Hermitian eigensystem decompositions by roughly a factor of two, via computing directly on a data matrix, rather than on the corresponding estimated correlation or covariance matrix. Although the SVD has long been utilized as a method of off-line or non-real-time computation, parallel computing architectures for its implementation in near real time have begun to emerge. The Generalized Singular Value Decomposition (GSVD) bears the same relationship to the computation of certain Hermitian generalized eigensystem decompositions that the ordinary SVD bears to the corresponding ordinary eigensystem decompositions. This paper discusses methods for computing the GSVD via a sequence of more familiar computations and indicates the relation of the GSVD to the MUSIC algorithm of R. Schmidt.