%Name Hans-Bj{\o}rn Foxby %Title: Duality and Equivalence %Abstract \newcommand{\Hom}{\mathrm{Hom}} For complexes $X$ and $\Omega$ of modules over a ring $R$ we consider the duality morphism $X\to\Hom_R(\Hom_R(X,\Omega),\Omega)$ together with the two equivalence morphisms $X\to\Hom_R(\Omega,\Omega\otimes_RX)$ and $X\leftarrow\Omega\otimes_R\Hom_R(\Omega,X)$ as well as their counterparts in the derived category. We present cases where these are isomorphisms including: Hartshorne's Affine Duality. Duality with respect to $R$ with connections to Auslander's G-dimension. Duality with respect to a stable Koszul complex. The corresponding equivalences.