In this chapter we study the general vector optimization problem \( \ \ \mathop {{{\,\textrm{WInf}\,}}}_{x\in X}\ \Phi (x,0_Z)\) associated with a perturbation mapping \(\Phi :X\times Z\rightarrow Y\cup \{+\infty _Y\}\) , where X, Y, Z are locally convex Hausdorff topological vector spaces and “WInf” stands for the weak infimum with respect to an ordering generated by a convex cone K (with \(\textrm{int} K \ne \emptyset \) ) in Y. Several representations of the epigraph of the conjugate mapping, \({{\,\textrm{epi}\,}}\Phi (.,0_Z)^*\) , are established, from which, variants of equivalent forms of the “vector inequality” \(\Phi (x,0_Z) \notin -{{\,\textrm{int}\,}}K, \ \mathrm{for\ all } \ x\in X \) (called vector Farkas lemmas) are proved. Armed with these basic tools, two kinds of dual problems of the mentioned problem with linear perturbations are proposed: The dual problem and the loose dual problem. Stable strong duality results between these pairs of primal-dual problems are proved. As applications of the results just obtained, we consider the cone-constrained vector problem. For such a class of problems, several forms of perturbation mappings \(\Phi \) are proposed which lead to the usual Lagrange dual problem, and several kinds of Fenchel-Lagrange dual problems together with stable strong duality results for these pairs of primal-dual problems. It is worth noticing that different perturbation mappings \(\Phi \) also give rise to corresponding versions of vector Farkas lemmas for vector systems defined by the mentioned problems (see Sects. 8 and 9). In the last section, Sect. 10, we consider the special case where \(Y= \mathbb {R}\) , for which, our results go back and in some cases extend the generalized Farkas lemmas, Lagrange, Fenchel-Lagrange duality results appeared in the literature in the last decades.