Algebraic cycles on abelian varieties - Application of abstract Fourier theory

We give a survey of the Fourier-Mukai transform on abelian varieties. This is a correspondence from an abelian variety to its dual abelian variety, constructed from the Poincare bundle. This correspondence was used by Lieberman and Mukai to compute cohomology and X-theory of abelian varieties and later by Beauville to study Chow groups of abelian varieties. We discuss the main theorem and the essential part of its proof (the so-called inversion formula) and as applications a theorem of Bloch on Pontryagin powers of algebraic cycles and the decomposition theorem of Beauville for Chow groups. We conclude by mentioning some further developments due to Deninger-Murre and Kunnemann.