This chapter extends the scalar equilibrium problem and develops existence results in Chap. 1 to the vector case. Section 3.1 presents some basic notions such as the algebraic interior and the vectorial closure of a set, cones, nonconvex separation functionals, and continuity of vector functions. Section 3.2 introduces a vector equilibrium problem \(\mathrm {(VEP}_{(f,C)})\) , concepts of solutions, relationships between kinds of solutions, and particular cases of \(\mathrm {(VEP}_{(f,C)})\) . Section 3.3 provides existence conditions for the kinds of solutions of \(\mathrm {(VEP}_{(f,C)})\) . The results are divided into two groups. The first one includes the results which are established based on the vector-proper quasimonotonicity. The second group consists of results formed based on the vector-cyclic quasimonotonicity. The last part of this section provides some forms of vectorial Ekeland variational principle. Section 3.4 introduces the vector quasiequilibrium problem, which is an extension of quasiequilibrium problem (QEP) to vector bifunctions.