  The paper studies quadratic and Koszul duality for modules over positively
graded categories. Typical examples are modules over a path algebra, which is
graded by the path length, of a not necessarily finite quiver with relations.
We present a very general definition of quadratic and Koszul duality functors
backed up by explicit examples. This generalises previous results in two
substantial ways: We work in the setup of graded categories, i.e. we allow
infinitely many idempotents and also define a ``Koszul'' duality functor for
not necessarily Koszul categories. As an illustration of the techniques we
reprove the Koszul duality of translation and Zuckerman functors for the
classical category O in a quite elementary and explicit way. As applications we
propose a definition of a "Koszul" dual category for integral blocks of
Harish-Chandra bimodules and for blocks outside the critical hyperplanes for
the Kac-Moody category O.
