We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szegorecurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel-Darboux formula. These results should be viewed as the direct analogue of the nearest neighbour recurrence relations from the theory of multiple orthogonal polynomials on the real line.