Let \(\omega_n (x) = \mu_n (x)/ \rho_l (x)\) \((n = 1,2,3,4)\) be the Bernstein-Szegö weights, where the \(\mu_k (x)\) are the classical Chebyshev weights, \(\mu_1 (x) = 1/ \sqrt {1 - x^2}\), \(\mu_2 (x) = \sqrt {1 - x^2}\), \(\mu_3 (x) = \sqrt {(1 - x)/(1 + x)}\), \(\mu_4 (x) = \sqrt {(1 + x)/(1 - x)}\), and \(\rho_l (x)\) is a positive polynomial of degree \(l\) on the interval \((-1,1)\). Four types of the Bernstein-Szegö polynomials orthogonal with respect to the weights \(\omega_n (x)\) are considered. For these systems of polynomials, among other topics, the following properties are investigated in detail: explicit forms, differential equations, three-term recurrence relations, extremal properties, and the Turán determinant. Equiconvergence between Bernstein-Szegö Fourier series and trigonometric Fourier series is obtained. Particular cases of Bernstein-Szegö polynomials are investigated separately.