We give the English translation of the paper ``Orthogonal systems of rational functions on the unit circle'' by M.M. Dzrbasian which appeared in two parts in Izvestiya Akademii Nauk Armyanskoi SSR, ser. Matematika 1 (1966) 3--24, 2 (1966) 106--125. This paper is one of the first to deal with orthogonal rational functions on the unit circle whose poles are fixed. These rational functions generalize the orthogonal polynomials of Szegö. We have tried to keep as close as possible to the original text, yet writing readable English.