Many real matching markets are subject to distributional constraints. To guide market designers faced with constraints, we propose new stability concepts. A matching is strongly stable if satisfying blocking pairs inevitably violates a constraint. We show that a strongly stable matching may not exist, and that existence is guaranteed if and only if all distributional constraints are trivial. To overcome this diffculty, we propose a more permissive concept, weak stability. We demonstrate a weakly stable matching always exists, implies e
ciency, and is characterized by standard normative axioms. These results are obtained in a more general environment than in existing studies, accommodating a wide variety of applications.