The concept of a characteristic function of a random variable or, equivalently, the Fourier transform of the law of a random variable, is a fundamental one in probability theory. It allows in an elegant and useful way to describe properties of measures and random variables. After presenting basic properties of the Fourier transform, the Riemann-Lebesgue lemma and the uniqueness theorem are proven. As applications, the computation of moments of measures, the characterization of Gaussian random vectors, and the independence of random variables are discussed.