In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution deals with a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function, this is achieved with the converse Lyapunov theorem dedicated to switched systems. In a second part, an explicit construction of a common control Lyapunov function is addressed to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.