In this paper, we consider the existence of radial solutions to a $k$-Hessian system in a general form. The existence of radial solutions is obtained under the assumptions that the nonlinearities in the given system satisfy $k$-superlinear, $k$-sublinear or $k$-asymptotically linear at the origin and infinity, respectively. The results presented in this paper generalize some known results. Examples are given for the illustration of the main results.