Asymptotic properties of Sobolev orthogonal polynomials

In this report we will survey some of the main ideas and tools which appeared recently in the study of the analytic properties of polynomials orthogonal with respect to inner products involving derivatives. Although some results on weak asymptotics are mentioned, the strong outer asymptotics constitutes the core of the paper. Both the discrete and the continuous cases are considered, and several open problems and conjectures are posed.