In this chapter, we consider sufficient conditions for (semi)continuity, Hölder/ Lipschitz continuity, and well-posedness of parametric vector equilibrium problems (VEPs), including their weak and strong variants. Two approaches are presented: the first is direct on the vector formulation and the second is via scalarization techniques. Regarding the first approach, for stability in the sense of semicontinuity, we apply cone continuity/semicontinuity properties to get upper semicontinuity results for solution maps of VEPs, while to obtain lower semicontinuity ones, generalized convexity assumptions are additionally needed. Finally, for continuity of these solution maps, we also aim at some results independent from the above ones about semicontinuity. The techniques used here are not very far from those in Chap. 5 , but the tools employed here are new. For stability in the Hölder/Lipschitz continuity or calmness sense, we resort to relaxed (quasi/pseudo) strong monotonicity and convexity of vector maps. When studying approximate solution maps, these properties are omitted or mitigated. Besides stability, a unified form of the Hadamard and Tikhonov well-posedness concepts in terms of semicontinuity, which is popular in the literature, is investigated as well in the first approach in this chapter. Moreover, we also establish relationships between well-posedness and stability of approximate solutions. Regarding the second approach (using scalarization), for weak VEPs we can use the linear scalarization by adding quasiconvexity hypotheses. But, for strong VEPs, we define many extensions of the Gerstewitz/Tammer nonlinear scalarization functions, and see as a byproduct that they can even be used also for weak VEPs. Furthermore, through these nonlinear scalarization techniques, we find interesting relationships between VEPs and set optimization problems, and then discuss stability for set optimization problems.