The role of (bi/multi/matrix) orthogonal polynomials in random matrices, integrable systems and combinatorics is well known. Our goal is to report on recent progress in the definition of suitable extensions of the notion of orthogonality where the polynomials are replaced by sections of appropriate line bundles on Riemann surfaces. We discuss their definition in the spirit of various generalizations of the Padé problem and the formulation of appropriate matrix Riemann Hilbert problems that allow to characterize them as well as control their asymptotic behaviour. Applications to Matrix Orthogonal Polynomials and the KP hierarchy will also be discussed.