The most common and therefore also most extensively studied non-positive curvature conditions for metric spaces are the ones due to Alexandrov on the one hand and due to Busemann on the other hand. Whereas the latter one describes the convexity of the distance function in a certain sense, the \(\text{CAT}(0)\)-condition can be viewed as a uniform convexity condition of the square of the distance function. In this paper the author weakens both these conditions in order to obtain what he refers to as \(k\)-convexity and \(L\)-convexity, respectively, noting that both, \(2\)-uniformly convex Banach spaces as well as \(\text{CAT}(1)\)-spaces with suitably bounded diameter, share both these properties. As both these concepts generalize important and well established geometric properties of metric spaces, it is only natural to as the question to which extent results obtained for the classical notions also hold for their generalizations. Among other interesting considerations, the author for instance proves a first variational formula for locally compact, locally geodesics extendable, geodesic metric spaces which are both, \(k\)- and \(L\)-convex. Furthermore, among several other geometric and analytic results, the author proves the solvability of the Dirichlet problem for maps into a wide class of metric spaces.