"Recent years have seen an increasing use of theoretically grounded market designs to solve important real-life matching problems. A key factor for these practical accomplishments is that matching theory has been successful in developing centralized mechanisms that are (1) stable, that is, give agents no incentive to contract outside the mechanism, and (2) strategy-proof for a significant subset of participants, that is, provide these economic agents with strong incentives to reveal their true preferences over available matching opportunities to the mechanism. This project aims to make significant progress towards solving two of the most important open questions in this research area.First, several high-profile practical applications have shown that there is scope for designing stable and strategy-proof mechanisms even when the weakest known sufficient conditions for the existence of such a mechanism are not satisfied. I aim to provide a complete characterization of the circumstances under which stability and strategy-proofness can be achieved by means of a centralized matching mechanism. Apart from providing a unifying explanation for all existing applications, the hoped for results can help to identify new real-world problems to which matching theory can be applied.Second, full centralization is not feasible for many important real-life matching markets. Decentralized markets often suffer from congestion, which induces strong incentives for participants to condition their behavior on available statistics about past market conditions. In the second part of this project, I plan to study the dynamic consequences of individual learning in decentralized matching markets using theoretical, empirical, and experimental methods. The key motivating application is a decentralized university admission system and my research has the potential to yield practically implementable advice on how to make such an admission process work more efficiently."