We study two-sided many-to-one matching markets with transferable utilities subject to distributional constraints on the set of feasible allocations. In such markets, we establish the efficiency of equilibrium arrangements and study conditions on the distributional constraints and agent preferences under which equilibria exist and can be computed efficiently. We first consider the setting when the number of institutions (e.g., firms in a labor market) is one and show that equilibrium arrangements exist irrespective of the constraint structure or agents' preferences. However, equilibrium arrangements may not exist in markets with multiple institutions even when agents have linear preferences. Thus, for markets with linear preferences, we study sufficient conditions on the constraint structure that guarantee the existence of equilibria using linear programming, which not only generalizes that of Shapley and Shubik (1971) to the many-to-one matching setting under distributional constraints but also provides a method to compute market equilibria efficiently.