This paper studies and solves the problem of asymptotic stabilization of switched systems consisting of unstable second-order linear time-invariant (LTI) subsystems. Necessary and sufficient conditions for asymptotic stabilizability are first obtained. If a switched system is asymptotically stabilizable, then the conic switching laws proposed in the paper are used to construct a switching law that asymptotically stabilizes the system. Switched systems consisting of two subsystems with unstable foci are studied first and then the results are extended to switched systems with unstable nodes and saddle points. The results are applicable to switched systems that consist of more than two subsystems.