This project offers a theoretical and empirical investigation of matching markets. Matching is, broadly speaking, the study of complementarities, which explains the formation of coalitions. Matching models are found in many applied fields within Economics: Labour Economics, Family Economics, Consumer theory of differentiated goods (hedonic models), Trade, etc. Desirable properties of these coalitions, such as stability, lead to testable implications of the surplus that individuals generate in a match, allowing for structural estimation of matching models.The goal of this proposal is to expand the frontiers of the theory of matching to design a very general and highly flexible model of matching that will lend itself to estimation and thus lead to empirical findings in various fields of Economics. Based on promising work initiated by the PI, this proposal seeks to bridge the gap between the theory and the empirics of matching markets that was traditionally observed in this literature.Particular focus will be given to situations where stable outcomes may not exist (such as unipartite, or one-to-many matching models), frictions, taxes. In these cases, a thorough investigation is carried on what solution concept should be used, and what are the testable implications.Applications will be given to various empirical issues or policy relevant questions such as:- The nature of the complementarities between senior and junior employees within teams,- The role played by the marriage market in the problem of rural depletion in China,- The impact of CEO risk aversion on assignment to firms, and on the CEO compensation package,- The pricing of attributes of French wines.