This comprehensive monograph presents the history and achievements of one of the most important figures in modern mathematics, covering the work of Fourier from his first memoir on the Analytical Theory of Heat to the latest developments in wavelet theory. Originally, Fourier series were used to describe and compute the functions which occur in heat diffusion and equilibrium, but they soon led to the development of new theories by Fourier's followers, and some of these original papers are considered in this text.