Koszul duality is a fundamental phenomenon in mathematics. In its original form, it gives a derived equivalence between the module category of a Koszul algebra A and its Koszul dual A!. Koszul algebras are quadratic, but the duality can be extended to non-homogeneous Koszul algebras U, i.e. filtered deformations of Koszul algebras. The Koszul dual in this case has the extra structure of a curved differential (cdg)-algebra. The derived category of the category of cdg-modules is not well-defined. In this talk, I will explain that it is still possible to obtain an equivalence between derived category of U and an explicit quotient of the homotopy category of injective modules over A!. If U has finite global dimension, this quotient is trivial. Examples of non-homogeneous Koszul algebras include algebras that are of interest in Representation Theory, like the Weyl algebra, the enveloping algebra of a Lie algebra, the symplectic reflection algebra and the deformed preprojective algebra of a quiver. This is joint work with Gwyn Bellamy and Isambard Goodbody. -- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.