Generalized exponents are among the simplest parabolic Kazhdan-Lusztig polynomials but yet there is no reasonably efficient way to compute them. The generalized exponents were defined and studied by Kostant in his paper on representations on polynomial rings long before Kazhdan-Lusztig theory existed: they are graded multiplicities of irreducible representations of compact groups in the space of harmonic polynomials on the corresponding Lie algebra. I will describe some known facts about them, including their relationship to classical exponents, and present a new approach for their computation which as far has been quite succesful.