This paper provides an overview of some of the important inverse problems in digital signal processing and describes some of the techniques currently used to solve them. Some of the inverse problems which are considered include deconvolution, noise removal, extrapolation and interpolation, and phase retrieval. Two classes of algorithms used to solve these problems are discussed: (1) the non-iterative solutions which construct an operator to apply to the observation and (2) the constrained iterative solutions which are based on the method of successive approximations.