  We introduce the notion of Mukai regularity (M-regularity) for coherent
sheaves on abelian varieties. The definition is based on the Fourier-Mukai
transform, and in a special case depending on the choice of a polarization it
parallels and strenghtens the usual Castelnuovo-Mumford regularity. Mukai
regularity has a large number of applications, ranging from basic properties of
linear series on abelian varieties and defining equations for their
subvarieties, to higher dimensional type statements and to a study of special
classes of vector bundles. Some of these applications are explained here, while
others make the subject of upcoming papers.
