Stone dualities allow to describe special classes of topological spaces
by means of (possibly finitary) partial orders. Typically, these
partial orders are given by the topology, a basis for it, or a
subbasis for it. The seminal result is the duality between the
categories of Stone spaces and that of Boolean algebras.
Other very important examples are the descriptions of
Scott domains as information systems and the
description of SFP domains as pre-locales.
It is worthwhile to mention also Martin-Lof's domain interpretation
of intuitionistic type theory.

Intersection types can be viewed also as a restriction of the domain
theory in logical form to the special case of
modeling pure lambda calculus by means of omega-algebraic complete
lattices. Intersection types have been used as a powerful tool both
for the analysis and the synthesis of lambda-models.  On the one hand,
intersection type disciplines provide finitary inductive definitions
of interpretation of lambda terms in models. On the other hand,
they are suggestive for the shape the domain model has to have in
order to exhibit certain properties.

More recently intersection (together with union) types  have been used
to build  fully abstract models of extensions of the lambda-calculus
including parallel features, of Higher-Order Processes, and of the 
\pi-calculus.