Early results on the emptiness of the core and the majority-rule-chaos results led to the recognition of the importance of modeling institutional details in political processes. A sample of the literature on game-theoretic models of political phenomena that ensued is presented. In the case of sophisticated voting over certain kinds of binary agendas, such as might occur in a legislative setting, equilibria exist and can be nicely characterized. Endogenous choice of the agenda can sometimes yield "sophisticated sincerity", where equilibrium voting behavior is indistinguishable from sincere voting. Under some conditions there exist agenda-independent outcomes. Various kinds of "structure-induced equilibria" are also discussed. Finally, the effect of various types of incomplete information is considered. Incomplete information of how the voters will behave leads to probabilistic voting models that typically yield utilitarian outcomes. Uncertainty among the voters over which is the preferred outcome yields the pivotal voting phenomenon, in which voters can glean information from the fact that they are pivotal. The implications of this phenomenon are illustrated by results on the Condorcet Jury problem, where voters have common interests but different information.