Is there a direct link between Poincare duality and Langlands duality, since both are reflected in functional equations?
http://mathoverflow.net/questions/73711/the-concept-of-duality
Functional analysis on the eve of the 21st century Contains article by Kapranov: Analogies between the Langlands Correspondence and Topological QFT. Follow-up here at MO.
Alex S mentioned some very general form of Poincare duality
Gross-Hopkins duality mentioned by Behrens here
Hartshorne: Residues and duality, LNM0020.
Something about string theory and a very general form of Poincare duality
An interesting framework by Greenlees, Dwyer, Iyengar
Duality for abelian schemes: See Oort: LNM0015.
Altman and Kleiman LNM0146 Intro to Grothendieck duality. Looks very readable.
Preprint in progress of Rognes: Topological arithmetic duality
Scholbach thesis hints at duality underlying L-function stuff.
Saito: Cohomological Hasse principle for a threefold over a finite field review. Uses “arithmetic Poincare duality”, according to the review.
http://nlab.mathforge.org/nlab/show/dualizable+object
Minhyong Kim gave a talk in Cambridge in Feb 2011 called “Remarks on non-abelian duality”
nLab page on Duality