In this survey, we present some of the more significant results in the literature on adverse selection in insurance markets. Sections 1 and 2 introduce the subject, and Sect. 3 discusses the monopoly model developed by Stiglitz (Rev Econ Stud 44:407–430, 1977) for the case of single-period contracts, which has been extended by many authors to the multi-period case. The introduction of multi-period contracts raises issues that are discussed in detail: time horizon, discounting, commitment of the parties, contract renegotiation, and accident underreporting. Section 4 covers the literature on competitive contracts, where the analysis is more complicated because insurance companies must take competitive pressures into account when they set incentive contracts. As pointed out by Rothschild and Stiglitz (Q J Econ 90:629–650, 1976), there is not necessarily a Nash equilibrium when there is adverse selection. However, market equilibrium can be sustained when principals anticipate competitive reactions to their behavior. Multi-period contracting is discussed. We show that different predictions on the evolution of insurer profits over time can be obtained from different assumptions concerning the sharing of information between insurers about an individual’s choice of contracts and accident experience. The roles of commitment and renegotiation between the parties to the contract are important. Section 5 introduces models that consider moral hazard and adverse selection simultaneously, and Sect. 6 covers adverse selection when people can choose their risk status. Section 7 discusses many extensions to the basic models such as risk categorization, multidimensional adverse selection, symmetric imperfect information, double-sided adverse selection, participating contracts, and nonexclusive contracting.