The Hilbert series of the graded ring associated to a projective variety contains a lot of information about the variety, {it e.g./}, dimension, degree, arithmetic genus etc. If the projective variety is nice then the Hilbert series can be (uniquely) writtenin the form $h(t)=f(t)/(1-t)^n$, where $f(t)$ is a polynomial with non-negative integer coefficients and $f(1) ot =0$.In this talk we consider the Hilbert series of determinantal varieties (including symmetric and skew-symmetric determinantal varieties). These varieties arise naturally in many branches of mathematics, {it e.g.}, in classical invariant theory. We give an interpretation of the coefficients of the numerator $f(t)$ of the Hilbert series as dimensions of representations of certain compact Lie groups. (The presented work is joint work with Enright and is an extension of previous work by Enright and Willenbring.)