Scholarship on the classification, origins, incentives, and consequences of mixed-member electoral systems has matured, especially over the last two decades. While mixed-member electoral systems (also known as mixed electoral systems) have been in constant use since Germany adopted a mixed-member proportional system for assembly elections following World War II, researchers did not begin to fully probe the implications of this electoral system until its expansion across the globe beginning in the 1990s. Mixed-member electoral systems share an important characteristic: voter preferences are translated into outcomes by at least two allocation formulas applied in the same election. While voters typically receive a ballot to select a representative in a constituency (often using first-past-the-post) and a ballot to select a party list (often using a form of proportional representation (PR)), the institutional features of mixed-member systems vary substantially. A crucial distinction among mixed-member systems is whether or not seat allocation in the constituency and proportional representation tiers is linked (mixed-member proportional, or MMP) or unlinked (mixed-member majoritarian, or MMM). Across the universe of mixed-member systems, one finds additional differences in the number of ballots voters receive; the electoral formulas and thresholds used to determine winners; the proportion of seats allocated to each component; the ability of candidates to contest seats in both components during the same election; and other critical aspects of the rules. Scholarship classifying mixed-member systems has highlighted different aspects of the rules to sort them into categories. A substantial amount of scholarship on mixed-member systems has emphasized the debate about the incentives that the systems generate. The “controlled comparison” approach treats the components as if they are independent from one another and the “contamination effects” approach treats the components as if they are interdependent. These competing schools of thought generate different expectations, with the former generally anticipating compliance with Duverger’s propositions and the latter anticipating divergence. Subsequent scholarship has been split about which approach better explains observed behavior. However, many of the perceived differences between the approaches may be artificial, generated by extreme interpretations of the theoretical expectations that lack appropriate nuance. In other words, it may be inappropriate to treat this scholarship as strictly dichotomous. The extant literature on mixed-member systems evaluates data from surveys, interviews, personnel files, roll-call voting, and election returns to understand the behavior of voters, candidates, parties, and legislators. It assesses how the incentives of mixed-member systems contribute to outcomes such as the party system, descriptive representation, and policy decisions. It also explores the presence or absence of a “mandate divide”: the expectation that members of parliament (MPs) selected in the constituency component might behave differently than their counterparts in the party list component. The research is often cross-national, but studies of certain countries with mixed-member systems predominate: Germany, Japan, and New Zealand among established democracies, and central or east European countries among transitional societies. The literature presents many opportunities to generate more nuanced theory, explore different research methodologies (e.g., experimental work), and extend spatial coverage to under-studied countries.