We consider circular ensembles with nonuniform weight functions. We investigate the universality of short-range and long-range level fluctuations, which are important in the study of quantum chaotic systems. We analyze a set of hierarchic relations among the correlation functions to obtain the level density for a wide class of potentials and to demonstrate universality of correlation functions in the case of weak periodic potentials (where the term potential refers to the logarithm of the weight function). Analytic study of circular unitary ensemble is done with the help of orthogonal polynomials on the unit circle. For circular orthogonal and symplectic ensembles, we introduce skew-orthogonal polynomials on the unit circle. We consider the asymptotic forms of the polynomials for the three types of ensembles with weak potentials to give a proof of the universality. The analytic results are verified by Monte Carlo simulations of the ensembles with different weight functions. We also discuss the implications of these results in the context of conductance fluctuations in mesoscopic systems and show that the universality breaks down for strong potentials.