The paper studies stability and stabilizability of Discrete event Dynamical Systems (DEDS's) in the framework of Ramadge and Wonham. We define stability and stabilizability in terms of the behavior of the DEDS's, i.e. the language generated by the state machines (SM's) used to model the DEDS. The notions of stability and stabilizability of DEDS's have been presented in a more general setting than in earlier works where they are defined in terms of legal and illegal states rather than languages. The notion of reversal of languages is used to provide algorithms for determining the stability and stabilizability of a given system.