There are many problems described by an inequality of a bifunction involving constraints that depend on the decision variable, such as quasioptimization problems, quasivariational inequalities, generalized Nash equilibria, traffic networks with elastic demands, etc. To deeply study these problems within a common framework, they are merged into a general problem, called quasiequilibrium problem. This chapter deals with the quasiequilibrium problem (QEP) and results on existence of its solutions. Section 2.1 introduces some basic notions and properties of semicontinuity for set-valued mappings which are necessary to state problems and to establish the existence results. Section 2.2 presents the definition of the quasiequilibrium problem and several related problems. Section 2.3 provides some fixed-point theorems for set-valued mappings which are used to establish existence results for (QEP). Section 2.4 presents conditions for the existence of solutions of (QEP). Section 2.5 discusses some set-valued versions of equilibrium/quasiequilibrium problems, where objective bifunctions are set-valued bifunctions. Section 2.6 contains applications of the results of Sects. 2.4–2.5 to several particular cases of (QEP) including Browder variational inclusions.