  This is a survey on selected developments in the theory of natural dualities
where the author had the opportunity to make with his foreign colleagues
several breakthroughs and move the theory forward. It is aimed as author's
reflection on his works on the natural dualities in Oxford and Melbourne over
the period of twenty years 1993-2012 (before his attention with the colleagues
in universal algebra and lattice theory has been fully focused on the theory of
canonical extensions and the theory of bilattices). It is also meant as a
remainder that the main problems of the theory of natural dualities,
Dualisability Problem and Decidability Problem for Dualisability, remain still
open.
  Theory of natural dualities is a general theory for quasi-varieties of
algebras that generalizes `classical' dualities such as Stone duality for
Boolean algebras, Pontryagin duality for abelian groups, Priestley duality for
distributive lattices, and Hofmann-Mislove-Stralka duality for semilattices. We
present a brief background of the theory and then illustrate its applications
on our study of Entailment Problem, Problem of Endodualisability versus
Endoprimality and then a famous Full versus Strong Problem with related
developments.
