The theory of tempered distributions provides the mathematical foundation for much of signal theory, especially DSP. Yet signals processing texts do not contain careful expositions of the theory, preferring instead to treat the subject as a collection of rules to be memorized and formally manipulated. Unfortunately, the manipulations are often performed in situations which cannot be justified by the theory, for example, in derivations of the sampling theorem. This article contains a concise, though mathematically rigorous introduction to the theory of tempered distributions. We subsequently apply distribution theory to give rigorous proofs of many of the basic results of signals analysis, including Whittaker's sampling theorem and the remarkable though lesser-known result that every periodic tempered distribution has a (generalized) Fourier series. The level of presentation is intentionally less abstract and the proofs contain more detail than one normally finds in purely mathematical treatments of distribution theory and will hopefully appeal to a wider audience. Published by Elsevier Science (USA).