In this paper, an SEIR epidemic model with nonlinear incidence is investigated. By applying the next generation matrix, the basic reproduction number determining whether the disease dies is found, and the existence of the equilibria of the model is discussed; according to the suitable Lyapunov function and the LaSalle invariance principle, it is proved that the disease free equilibrium is globally asymptotically stable as the basic reproduction number is less than or equal to unity; by means of the Lyapunov direct method, it is testified that the endemic equilibrium is globally asymptotically stable as the basic reproduction number is greater than unity. Finally, the theoretical results obtained here are verified by numerical simulations for the SEIR model with a specific incidence.