Structural features of circulant matrices are utilized to study stabilities of circulant systems in this paper. They are that a necessary and sufficient condition for asymptotic stability of circulant systems is given; when systems contain uncertain circulant parameters, a robust stable region is determined. Besides, we present a switched circulant system model and a nonlinear circulant system model. For the former, we explore a necessary and sufficient condition for asymptotic stability under arbitrary switching laws with a common quadratic Lyapunov function constructed, and for the latter, we find out a sufficient condition for locally asymptotically stability at the origin. Finally, simulation examples illustrate the main results of this paper.