In this paper, we deal with vectoroptimization problems where the images of the objective maps aregiven in linear spaces not endowed with any topologicalstructure. Firstly, we propose new concepts of semicontinuity ofvector-valued maps acting from metric spaces into linearones and discuss their properties and characterizations. Next, using Zorn’s Lemma and these properties, we formulateexistence conditions for such problems. Then, we introduce two notions of Levitin-Polyak well-posedness for thereference problems, and study sufficient conditions for theirfulfillment, as well as relationships of these notions. Finally, using theKuratowski measure of noncompactness, several metric characterizations of such well-posedness properties for vectoroptimization problems are investigated. Many examples are given to analyze our results.