We introduce multiple orthogonal polynomials on the unit circle. We show how this is related to simultaneous rational approximation to Carathéodory functions (two-point Hermite–Padé approximation near zero and near infinity). We give a Riemann–Hilbert problem for which the solution is in terms of type I and type II multiple orthogonal polynomials on the unit circle, and recurrence relations are obtained
from this Riemann–Hilbert problem. Some examples are given to give an idea of the behavior of the zeros of type II multiple orthogonal polynomials.