Let H be a self-adjoint operator defined on an infinite dimensional Hilbert space. Given some
spectral information about H, such as the continuity of its spectral measure, what can be said about
the asymptotic spectral properties of its finite dimensional approximations? This is a natural (and
general) question, and can be used to frame many specific problems such as the asymptotics of zeros of
orthogonal polynomials, or eigenvalues of random matrices. We shall discuss some old and new results
in the context of this general framework and present various open problems.