This book, written by one of the world's leading experts in the area, includes the homogeneous and nonhomogeneous quadratic duality theory over a nonsemisimple, noncommutative base ring, the Poincare–Birkhoff–Witt theorem generalized to this context, and triangulated equivalences between suitable exotic derived categories of modules, curved DG comodules, and curved DG contramodules. The thematic example, meaning the classical duality between the ring of differential operators and the de Rham DG algebra of differential forms, involves some of the most important objects of study in the contemporary algebraic and differential geometry. For the first timein the history of Koszul duality the derived D-\Omega duality is included into a general framework. Examples highly relevant for algebraic and differential geometry are discussed in detail.