Motivated by recent applications in control theory, we study the feedback stabilizability of switched systems, where one is allowed to chose the switching signal as a function of x(t) in order to stabilize the system. We propose new algorithms and analyze several mathematical features of the problem which were unnoticed up to now, to our knowledge. We prove complexity results, (in) equivalence between various notions of stabilizability, existence of Lyapunov functions, and we provide a case study for a paradigmatic example introduced by Stanford and Urbano in the 1990s.