A constraint satisfaction problem for structure $B$ has a duality if the
existence of a homomorphism from a given structure $A$ to $B$ is equivalent
to the non-existence of a homomorphism to $A$ from a structure belonging to
a certain well-behaved class. This talks surveys results concerning finite duality,
bounded pathwidth duality, and bounded treewidth duality, where the above
mentiooned class of ``obstructions has the corresponding property.
We discuss the following aspects for structures with a given duality: definability
of the corresponding CSP in logics, sufficient algebraic conditons, and the
complexity of the meta-problem (of recognizing structures with the given duality).