Summary: Distributional constraints are important in many market design settings. Prominent examples include the minimum manning requirements at each Army branch in military cadet matching and diversity considerations in school choice, whereby school districts impose constraints on the demographic distribution of students at each school. Standard assignment mechanisms implemented in practice are unable to accommodate these constraints. This leads policymakers to resort to ad hoc solutions that eliminate blocks of seats ex ante (before agents submit their preferences) to ensure that all constraints are satisfied ex post (after the mechanism is run). We show that these current solutions ignore important information contained in the submitted preferences, resulting in avoidable inefficiency. We then introduce new dynamic quotas mechanisms that result in Pareto superior allocations while at the same time respecting all distributional constraints and satisfying important fairness and incentive properties. We expect the use of our mechanisms to improve the performance of matching markets with distributional constraints in the field.