First we give necessary and sufficient conditions on a set of intervalsE l =∪ l j=1 [ϕ 2j−1 , 2j ], 1 <…< 2l and 2l −ϕ 1 ⩽2π, such that onE l there exists a real trigonometric polynomialτ N (ϕ) with maximal number, i.e.,N+l, of extremal points onE l . The associated algebraic polynomialT N (z)=z N/2 τ N (z),z=e iϕ , is called the complex Chebyshev polynomial. Then it is shown that polynomials orthogonal onE l have periodic reflection coefficients if and only if they are orthogonal onE l with respect to a measure of the form[formula]certain point measures, whereAis a real trigonometric polynomial with no zeros onE l and there exists a complex Chebyshev polynomial onE l . Let us point out in this connection that Geronimus has shown that orthogonal polynomials generated by periodic reflection coefficients of absolute value less than 1 are orthogonal with respect to a measure of the above type. Furthermore, we derive explicit representations of the corresponding orthogonal polynomials with the help of the complex Chebyshev polynomials. Finally, we provide a characterization of those definite functionals to which orthogonal polynomials with periodic reflection coefficients of modulus unequal to one are orthogonal.