The discrete Fourier transformation is one of the most important tools in the analysis of functions and signals, but its detailed aspects are often disregarded. Fourier and sampling theorems, Parseval’s equality, and power spectral densities are introduced, and the concepts of signal uncertainty, aliasing, and leakage are discussed. Sparse and non-uniform Fourier transformations are introduced. Transformations with orthogonal polynomials (Legendre, Chebyshev, Laguerre, Hermite), as well as the Chebyshev TL and TB functions are described. Their relation to quadrature formulas is explained. The numerical computation of the Laplace transformation and its inverse is introduced in the context of differential equations for transfer functions of physical systems. The continuous and discrete Hilbert transformations are discussed next, introducing the concept of the analytic signal, and their relation to the Kramers–Krönig dispersion relations is given. Finally, we describe the continuous and discrete wavelet transformations. The Examples and Problems include multiplication of polynomials by FFT, Fourier analysis of realistic acoustic signals and the Doppler effect, and the analysis of brain potentials by the Hilbert transform.