This paper reviews the use of Fourier transform methods in the pricing of contingent claims. This is a very promosing topic in finance, given the scarcity of closed-form solutions for derivative prices. It is shown that solving for the Fourier Transform is much easier than solving for the price, especially so under complex probability models, such as affine jump diffusions. In fact, explicit solutions for the Fourier transform are guaranteed for many types of contingent claims. As a consequence, the only remaining numerical issue in that case is the inversion of the Fourier transform.