For a quantale V we introduce V-approach spaces via V-valued point-set-distance functions and, when V is completely distributive, characterize them in terms of both, so-called closure towers and ultrafilter convergence relations. When V is the two-element chain 2, the extended real half-line [0, ∞], or the quantale ∆ of distance distribution functions, the general setting produces known and new results on topological spaces, approach spaces, and the only recently considered probabilistic approach spaces, as well as on their functorial interactions with each other.Keywords: quantale, V-closure space, V-approach space, discrete V-presheaf monad, lax distributive law, lax (λ, V)-algebra, probabilistic approach space, algebraic functor, change-of-base functor.Throughout the paper, let V = (V, ⊗, k) be a (unital but not necessarily commutative) quantale, i.e., a complete lattice with a monoid structure whose binary operation ⊗ preserves suprema in each variable. There are no additional provisions for the tensor-neutral element k vis-à-vis the bottom and top elements in V, i.e., we exclude neither the case k = ⊥ (so that |V| = 1), nor k < ⊤. The V-powerset functor P V : Set / / Set is given by