Abstract: Ratner's theorems on unipotent actions on e.g. the quotient SL(n,Z)\SL(n,R) have found many interesting applications in different areas of mathemtaics, most notably in number theory. Furstenberg, Katok, Spatzier, and Margulis have conjectured that similar results should hold for the action of the full diagonal subgroup once n is at least 3. We will discuss partial results towards a classification of invariant measures and some of its applications.