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This thesis develops a framework to study switching stabilization problems of continuous-time switched systems. Four types of switching stabilizability are formalized under different assumptions on the switching control input and adopted solution notions. On the time-domain, we define switching stabilizability as the existence of a measurable switching signal under which the resulting time-varying system is asymptotically stable. Discrete switching stabilizability is defined similarly but requires the switching signal to be piecewise constant on intervals of uniform length. On the state-feedback side, we define feedback stabilizability in Filippov sense (respectively, sample-and-hold sense) as the existence of a feedback law under which closed-loop Filippov solution (respectively, sample-and-hold solution) is asymptotically stable. The study on switching stabilization is divided into the special switched linear system case and the general case. For switched linear systems, it is proved that the four switching stabilizability notions are equivalent and their sufficient and necessary condition is the existence of a piecewise quadratic control-Lyapunov function that can be expressed as the pointwise minimum of a finite number of quadratic functions. For the general switched nonlinear systems, we focus on the notion of feedback stabilizability in Filippov sense. A piecewise smooth control-Lyapunov function framework is developed for constructive design of switching laws and stability guarantee for sliding motions. We discuss various technical issues emerging from nonsmooth surfaces of control-Lyapunov function and discontinuous surfaces of vector field. Sufficient conditions are derived for closed-loop stability including sliding motions on the attractive sliding surfaces.