The aim of this paper is to present theorems on the exitence of zeros for mappings defined on convex subsets of topological vector spaces with values in a vector space. In addition to natural assumptions of continuity, convexity, and compactness, the mappings are subject to some geometric conditions. In the first theorem, the mapping satisfies a "Darboux-type" property expressed in terms of an auxiliary numerical function. Typically, this functions is, in this case, related to an order structure on the target space. We derive an existence theorem for "obtuse" quasiconvex mappings with values in an ordered vector space. In the second theorem, we prove the existence of a "common zero" for an arbitrary (not necessarily countable) family of mappings satisfying a general "inwardness" condition againg expressed in terms of numerical functions (these numerical functions could be duality pairings (more generally, bilinear forms)). Our inwardness condition encompasses classical inwardness conditions of Leray-Schauder, Altman, or Bergman-Halpern types.