A matching market assigns objects to individuals, or individuals to each other. Typically, the different objects are indivisible, and individuals differ in how much they value each of them, so that the assignment has important implications for the well-being of the individuals. Moreover, relevant applications involve markets where the use of monetary payments is limited or infeasible, such as public school choice, assignment of graduate students, or the exchange of live-donor kidneys for transplantation. In these markets, exhausting all opportunities for mutually beneficial exchange with the limited means available is important for the well-being and, in the case of the last example, the health of those involved. This chapter will demonstrate how economic theory can offer some guidance for the design of ...