We study two-sided matching markets with asymmetric information, and define a concept of stability as the consistency of matching outcomes and the uninformed players' beliefs: there are no profitable pairwise deviations, in the expected utility sense, from the matching outcomes given the beliefs, and the beliefs are endogenously defined as a component of the stability notion. 
We define a new notion of efficiency relative to information revealed by the observable stable matching outcomes, and identify environments in which all stable matchings are efficient. We also study and discuss further extensions and applications of the theory, such as the core, correlated beliefs, two-sided uncertainty, and refinements.