The relation between (standard) orthogonal polynomials and random matrix theory is, by now, somewhat classical. In rough terms, we can express the partition function (and many other related quantities) of random matrix models in terms of orthogonal polynomials, so that asymptotic questions in random matrix theory can be immediately translated to the asymptotic theory of orthogonal polynomials. A natural generalization of standard orthogonality on the real line is provided by non-Hermitian orthogonality, where the standard bona fide inner product is replaced by orthogonality with respect to a non-Hermitian bilinear form, typically expressed in terms of contour integrals in the complex plane. Although the direct connection to random matrix theory is lost, the formal partition function associated to the model is meaningful to enumeration problems of graphs in compact Riemann surfaces of arbitrary genus. Hence the asymptotic theory of these non-Hermitian orthogonal polynomials is still of interest. However, due to the analytic character of the integrands defining the orthogonality there is a lot of freedom in the choice of contour of integration for the orthogonality, and consequently classical potential-theoretic techniques have to be suitably adapted and improved in combination with more recent Riemann-Hilbert methods. In the first part of our talk, we will survey some old and not-so-old results on the asymptotic theory of these non-Hermitian orthogonal polynomials, focusing on how their asymptotics can be extracted with the aid of the S-curves. This first part is partially based on joint work with Arno Kuijlaars (KU Leuven - Belgium). But this story is not yet over! Another generalization of standard or- thogonality is provided by the multiple orthogonality, where the conditions of orthogonality are split into two (or more) measures. This generalization is again physically meaningful: many random matrix models (and also random path models) can be described in terms of such multiple orthogonal polynomials. However, the asymptotic analysis of such polynomials has so far been restricted to situations involving symmetries in the model. So in the second part of our talk we plan to discuss some more recent developments towards removing such symmetry constraints. This is an ongoing project with Andrei Martinez-Finkelshtein (Universidad de Almeria - Spain), and as such the whole picture is not yet complete. At any rate we will talk about how the S-contour business can be generalized to this situation, and how a salad of critical measures and quadratic differentials comes to the table. Speaker(s): Guilherme Silva (UM)