Many real matching markets are subject to distributional constraints. These constraints often take the form of restrictions on the numbers of agents on one side of the market matched to certain subsets on the other side. Real-life examples include restrictions on regions in medical matching, academic master's programs in graduate admission, and state-financed seats for college admission. Motivated by these markets, we study design of matching mechanisms under distributional constraints. We show that existing matching mechanisms suffer from inefficiency and instability, and propose a mechanism that is better in terms of efficiency, stability, and incentives while respecting the distributional constraints. (JEL C70, D61, D63)