Harmonic analysis on reductive symmetric spaces

We give a relatively non-technical survey of some recent advances in the Fourier theory for semisimple symmetric spaces. There are three major results: An inversion formula for the Fourier transform, a Paley-Wiener theorem, which describes the Fourier image of the space of compactly supported smooth functions, and the Plancherel theorem, which describes the decomposition into irreducibles of the regular representation.