will first review the classical picture for GL(n), where objects from geometry (intersection theory of Grassmannians) and representation theory (invariant theory and eigenvalues of sums of Hermitian matrices) are related in many ways (by the work of ancient authors, Klyachko, Totaro, Knutson-Tao ...). I will then talk about my geometric proofs of some of these results, and the "geometrization" of the numerical relations between intersection theory and invariant theory. Parts of the geometric picture obtained above in the GL(n) case generalises (with surprises) to arbitrary Lie groups. I will summarize the recent progress made by using these methods for arbitrary groups (which is joint work with Shrawan Kumar).