  In this paper, we solve the long standing open problem on exact
dimensionality of self-affine measures on the plane. We show that every
self-affine measure on the plane is exact dimensional regardless of the choice
of the defining iterated function system. In higher dimensions, under certain
assumptions, we prove that self-affine and quasi self-affine measures are exact
dimensional. In both cases, the measures satisfy the Ledrappier-Young formula.
