Fractional-order derivatives provide a powerful instrument for describing the memory and hereditary of different substances. A growing number of works from various fields throughout science and engineering deal with dynamical systems expressed by fractional-order equations. In this paper, the existence and uniqueness of the solution to the initial problem for systems of fractional differential equations are derived. Based on the Schur theorem, the existence and uniqueness of the local solution to the linear system are given. Furthermore, the effectiveness of the proposed method is verified by a simulated example. For a nonlinear system, the existence and uniqueness of the solution are obtained by the Schauder fixed point theorem and the Banach fixed point theorem, respectively.