In this survey article, some linear transformations that play a fundamental role in signal processing and optical systems are reviewed. After a brief discussion of the general theory of linear systems, specific linear transformations are introduced. An important class of signals to which most of these linear transformations are applied is the class of bandlimited signals and some of its generalizations. The article begins by an introduction to this class of signals and some of its properties, in particular, the property that a bandlimited signal can be perfectly reconstructed from its samples on a discrete set of points. The main tool for the reconstruction is known as the sampling theorem. Some of the transformations presented, such as the windowed Fourier transform, the continuous wavelet transform, the Wigner distribution function, the radar ambiguity function, and the ambiguity transformation, fall into the category of time-frequency, scale-translation, or phase-space representations. Such transformations make it possible to study physical systems from two different perspectives simultaneously. Another group of transformations presented is closely related to the Fourier transform, such as the fractional Fourier transform and some of its generalizations. Chief among these generalizations are the linear canonical transform and the special affine Fourier transform. Properties and applications of these transforms in optical systems, together with sampling theorems for signals bandlimited in the domains of the aforementioned transforms, are introduced.