This paper contains first steps towards a Szego theory of orthogonal rational matrix-valued functions on the unit circle T. Hereby we are guided by former work of Bultheel, Gonzalez-Vera, Hendriksen, and Njastad on scalar orthogonal rational functions on the one side and by investigations of Delsarte, Genin, and Kamp on orthogonal matrix polynomials on the other side. An essential characteristic of our matricial orthogonalization procedure is marked by an intensive interplay between left and right matrix-valued inner products generated by a nonnegative Hermitian Borel measure on the unit circle. The main feature of our approach is the distinguished role of Christoffel-Darboux formulas. We consider pairs of rational matrix-valued functions linked via Christoffel-Darboux type relations as an own subject. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.