  We present an abstract unifying framework for interpreting Stone-type
dualities; several known dualities are seen to be instances of just one
topos-theoretic phenomenon, and new dualities are introduced. In fact,
infinitely many new dualities between preordered structures and locales or
topological spaces can be generated through our topos-theoretic machinery in a
uniform way. We then apply our topos-theoretic interpetation to obtain results
connecting properties of preorders and properties of the corresponding locales
or topological spaces, and we establish adjunctions between various kinds of
categories as natural applications of our general methodology. In the last part
of the paper, we exploit the theory developed in the previous parts to obtain a
topos-theoretic interpretation of the problem of finding explicit descriptions
of models of 'ordered algebraic theories' presented by generators and
relations, and give several examples which illustrate the effectiveness of our
methodology. In passing, we provide a number of other applications of our
theory to Algebra, Topology and Logic.
