A quasiordered set Q (qoset for short) is ideal-distributive iff Q I , the lattice of ideals in the sense of Frink, is distributive. The principal ideal embedding of Q in Q I is characterized by certain density properties, by extremal conditions and by a universal property. The reflector I from the category of qosets and ideal-continuous maps (where inverse images of ideals are ideals) to the category of algebraic lattices and join-preserving maps has several interesting subreflectors, for example, from the category of certain ideal-distributive qosets (including all bounded distributive lattices) to the subcategory of algebraic frames. Generalizing the classical Stone duality for distributive (semi-)lattices, we establish a dual equivalence between the category of ideal-distributive posets with so-called ∧-stable ideal-continuous maps and the category of pairs ( X, B ) where B is a base of a sober topology on X and B is meet-dense in the collection of all compact open sets; morphisms in this category preserve the distinguished bases under inverse images. We also study a self-dual notion of distributivity for qosets, compare it with ideal-distributivity and determine the corresponding Stone representation.