This paper studies switching stabilization problems for general switched nonlinear systems. A piecewise smooth control-Lyapunov function (PSCLF) approach is proposed and a constructive way to design stabilizing switching laws is developed. The switching law is constructed via directional derivative of the PSCLF with a careful discussion on various technical issues that may occur on the nonsmooth surface. Sufficient conditions are derived to ensure stability of closed-loop Filippov solutions including possible sliding motions. The proposed PSCLF approach provides a unified framework to study switching stabilization problems with a systematic consideration of sliding motions. Application to stabilization of switched nonlinear systems is illustrated through a numerical example. (C) 2016 Elsevier Ltd. All rights reserved.