abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Given a homomorphism $f$ of schemes, one says that it satisfies Grothendieck duality if the (derived) direct image functor $f_\ast$ on quasicoherent sheaves has a (derived) right adjoint $f^!$. This is Verdier duality in a “Grothendieck context” of six operations.
Suppose $f\colon X \to Y$ is a quasi-compact and quasi-separated morphism of schemes; then the triangulated functor $\mathbf{R}f_*\colon D_{qc}(X)\to D(Y)$ has a bounded below right adjoint. In other words, $\mathbf{R}Hom_X(\mathcal{F}, f^\times \mathcal{G})\stackrel{\sim}{\to} \mathbf{R}Hom_Y(\mathbf{R}f_*\mathcal{F}, \mathcal{G})$ is a natural isomorphism.
Robin Hartshorne, Residues and duality (Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne.) Springer LNM 20, 1966 MR222093
Domingo Toledo, Yue Lin L. Tong, Duality and intersection theory in complex manifolds. I., Math. Ann. 237 (1978), no. 1, 41–77, MR80d:32008, doi
Mitya Boyarchenko, Vladimir Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier, arxiv/1108.6020
Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les $\mathcal{D}_X$-modules cohérents, Travaux en Cours 35. Hermann, Paris, 1989. x+254 pp. MR90m:32026
Amnon Neeman, Derived categories and Grothendieck duality, in: Triangulated categories, 290–350, London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press 2010
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236, MR96c:18006, doi
Brian Conrad, Grothendieck duality and base change, Springer Lec. Notes Math. 1750 (2000) vi+296 pp.
Joseph Lipman, Notes on derived functors and Grothendieck dualitym in: Foundations of Grothendieck duality for diagrams of schemes, 1–259, Lecture Notes in Math. 1960, Springer 2009, doi, draft pdf
J. Lipman, Grothendieck operations and coherence in categories, conference slides, 2009, pdf
Alonso Tarrío, Leovigildo; Jeremías López, Ana; Joseph Lipman, Studies in duality on Noetherian formal schemes and non-Noetherian ordinary schemes, Contemporary Mathematics 244 Amer. Math. Soc. 1999. x+126L. MR2000h:14017; Duality and flat base change on formal schemes, Contemporary Math. 244 (1999), pp. 3–90.
J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I., Astérisque No. 314 (2007), x+466 pp. (2008) MR2009h:14032; II. Astérisque No. 315 (2007), vi+364 pp. (2008) MR2009m:14007; also a file at K-theory archive THESE.pdf
Amnon Yekutieli, James Zhang, Rigid dualizing complexes over commutative rings, Algebr. Represent. Theory 12 (2009), no. 1, 19–52, doi
Amnon Yekutieli, The residue complex of a noncommutative graded algebra, J. Algebra 186 (1996), no. 2, 522–543; Smooth formal embeddings and the residue complex, Canad. J. Math. 50 (1998), no. 4, 863–896, MR99i:14004; Rigid dualizing complexes via differential graded algebras (survey), in: Triangulated categories, 452–463, London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press 2010, MR2011h:18015
Roy Joshua, Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories (pdf)