This paper addresses backstepping controller synthesis for piecewise affine (PWA) systems. The main contribution of the paper is to formulate controller design for a large class of PWA systems as a convex problem. Integrator backstepping is proposed as the principal design step in constructing Lyapunov functions for PWA systems in strict feedback form. The controller synthesis problem is divided into two cases. The first case consists of the construction of a sum of squares (SOS) Lyapunov function for PWA systems with discontinuous vector fields. The second case addresses the construction of a piecewise polynomial Lyapunov function for PWA systems with continuous vector fields. After constructing a (piecewise) polynomial Lyapunov function, controller synthesis for a PWA system can be formulated as an SOS program, which is a convex optimization problem and can be efficiently solved. The new synthesis method is applied to a numerical example.