The singular value decomposition has a number of applications in digital signal processing. However, the the decomposition must be computed from a matrix consisting of both signal and noise. It is therefore important to be able to assess the effects of the noise on the singular values and singular vectors,---,a problem in classical perturbation theory. In this paper we survey the perturbation theory of the singular value decomposition. (Also cross-referenced as UMIACS-TR-90-124) Appeared in SVD and Signal Processing, II, R. J. Vacarro ed., Elsevier, Amsterdam, 1991.