There exist discontinuous functions that cannot be represented by power series. However, powers are not the unique type of known functions: there are also the trigonometric and hyperbolic functions, studied in basic mathematics, and the special functions. These functions can also be used to express other functions, just as in the case of powers. Among the series thus generated we have the Fourier series, Fourier–Bessel series, and Fourier–Legendre series, which are discussed here. For the Fourier series, we present some classical results such as orthogonality relations and the Parseval identity. For Fourier–Bessel series and Fourier–Legendre series, orthogonality plays an importante rule because it is related with the boundary conditions, respectively, Dirichlet conditions and Neumann conditions. Some solved exercises are discussed step by step, e.g. in the study of the two-dimensional Laplace equations in cartesian coordinates. As applications we evaluate some infinite sums. We conclude with a list of proposed exercises; among them, some interesting applications involving Legendre polynomials and Bessel functions are left to the reader, all of them with the correspondint answer and/or suggestion.