This paper is concerned with the group-theoretical background of integral transforms and the role they play in signal processing on the sphere. An overview of topological groups, measure and representation theories, and an overview of the Windowed Fourier Transform and the Continuous Wavelet Transform are presented. A group-theoretical framework within which these transforms arise is presented. The connection between integral transforms and square-integrable group representations is explored. The theory is also generalized beyond groups to homogeneous spaces. The abstract theory is then applied to signal processing on the sphere with a discussion of both global and local methods. A detailed derivation of the continuous spherical harmonic transform is presented. Global methods such as the spherical Fourier transform and convolution are presented in an appendix as well as some background material from group theory and topology.