The term duality can refer to many notions. Most of them are instances of the notion of dual category?. Often a duality is formulated in terms of cohomology?.
dual category?
dual vector space?
Poincaré duality? for finite dimensional (oriented) closed manifolds
Pontryagin duality? for commutative (Hausdorff?) topological groups?
Cartier duality of a finite flat commutative group scheme?
Serre duality? on nonsingular projective algebraic varieties which has as a special case the statement of the Rieman-Roch theorem?
Grothendieck duality?, coherent duality? for coherent sheaves?
Verdier duality? for abelian categories of sheaves; e.g. for a category of sheaves of abelian groups.
Artin-Verdier duality? generalizing Tate duality? for constructible sheaves over the spectrum of a ring of algebraic numbers?.