We study private-good allocation under general constraints. Several prominent examples are special cases, including house allocation, roommate matching, social choice, and multiple assignment. Every individually strategy-proof and Pareto efficient two-agent mechanism is an "adapted local dictatorship." Every group strategy-proof N-agent mechanism has two-agent marginal mechanisms that are adapted local dictatorships. These results yield new characterizations and unifying insights for known characterizations. We find all group strategy-proof and Pareto efficient mechanisms for the roommates problem. We give a related result for multiple assignment. We prove the Gibbard--Satterthwaite Theorem and give a partial converse.