  We review some recent results on asymptotic properties of polynomials of
large degree, of general holomorphic sections of high powers of positive line
bundles over Kahler manifolds, and of Laplace eigenfunctions of large
eigenvalue on compact Riemannian manifolds. We describe statistical patterns in
the zeros, critical points and L^p norms of random polynomials and holomorphic
sections, and the influence of the Newton polytope on these patterns. For
eigenfunctions, we discuss L^p norms and mass concentration of individual
eigenfunctions and their relation to dynamics of the geodesic flow.
