  Multiple orthogonal polynomials are a generalization of orthogonal
polynomials in which the orthogonality is distributed among a number of
orthogonality weights. They appear in random matrix theory in the form of
special determinantal point processes that are called multiple orthogonal
polynomial (MOP) ensembles. The correlation kernel in such an ensemble is
expressed in terms of the solution of a Riemann-Hilbert problem, that is of
size (r+1) x (r+1) in the case of r weights. A number of models give rise to a
MOP ensemble, and we discuss recent results on models of non-intersecting
Brownian motions, Hermitian random matrices with external source, and the two
matrix model. A novel feature in the asymptotic analysis of the latter two
models is a vector equilibrium problem for two or more measures, that describes
the limiting mean eigenvalue density. The vector equilibrium problems involve
both an external field and an upper constraint.
