Multiple orthogonal polynomials and strong Szego asymptotics

We give a survey of results on the asymptotic theory of multiple orthogonal polynomials. We present results that are already classical and completely new ones. The focus is on the description of methods for the derivation of these results and on their applications, classical and contemporary. This main aim is to describe the strong asymptotic behaviour of multiple orthogonal polynomials with respect to a common system of weights satisfying the Szegő condition. Bibliography: 123 titles.