We study vector optimization problems with solid non-polyhedral convex ordering cones, without assuming any convexity or quasiconvexity assumption. We state a Weierstrass-type theorem and existence results for weak efficient solutions for coercive and noncoercive problems. Our approach is based on a new coercivity notion for vector-valued functions, two realizations of the Gerstewitz scalarization function, asymptotic analysis and a regularization of the objective function. We define new boundedness and lower semicontinuity properties for vector-valued functions and study their properties. These new tools rely heavily on the solidness of the ordering cone through the notion of colevel and level sets. As a consequence of this approach, we improve various existence results from the literature, since weaker assumptions are required.