This chapter deals with the scalar equilibrium problem (EP) and the existence of solutions to this problem systematically. The main points of this chapter are as follows. In Sect. 1.1 we state problem (EP) and clarify the relationships between (EP) and some other problems. This section also provides some basic concepts and results such as semicontinuity of scalar functions, Sperner’s lemma, and Brouwer’s fixed-point theorem. Section 1.2 presents existence conditions for solutions of (EP) under convexity assumptions. Here, we also systematize several notions closely related to the convexity structure such as hemicontinuity and sign continuity of bifunctions, generalized convexity of scalar functions, and monotonicity properties of bifunctions. Section 1.3 provides existence results in nonconvex settings under pure topological assumptions. The results are established for certain classes of bifunctions, including the class of bifunctions satisfying the triangle inequality, the class of generalized cyclically monotone bifunctions, the class of KKM-bifunctions, and the one of connectedness-bifunctions.