I've spent many years researching Koszul duality in its various versions.  To me, Koszul duality is a fundamental homological phenomenon which has many manifestations, e.g.

1. the relation between the homotopy groups of a topological space and its (co)homology groups;
2. the relation between an augmented algebra A and its Ext-algebra $\text{Ext}_{A}(k,k)$ (and between modules over these);
3. the relation between the ring of differential operators and the de Rham DG-algebra of differential forms (and between modules over these).

Version 1. is obviously complicated and one can approach it from various angles, like rational homotopy theory or stable homotopy theory.  The former is related to the duality between commutative and Lie DG-algebras (Quillen), while the latter is more akin to module duality, in that it is additive.  Morphisms from the sphere spectrum to the Eilenberg-MacLane spectrum are very simple to describe, while endomorphisms of the latter spectrum are more complicated and endomorphisms of the former one are much more complicated.  One can view the sphere spectrum as a kind of projective generator and the Eilenberg-MacLane spectrum as a kind of irreducible module.

Version 2. is better to approach by splitting it in two branches, namely
2a. the relation between a conilpotent coalgebra $C$ and its Ext-algebra $\text{Ext}_{C}(k,k)$; and
2b. the relation between an augmented algebra $A$ and its Tor-coalgebra $\text{Tor}^{A}(k,k)$.  Then one can generalize 2a. to DG-coalgebras, and 2b. to DG-algebras, which makes these two correspondences inverse to each other.  Subsequently one can notice that the augmentation assumption or structure is largely irrelevant for 2b., and generalize this duality even further, obtaining a correspondence between conilpotent CDG- (curved DG-) coalgebras and (nonaugmented) DG-algebras.

Version 3. is a relative one, with the ring of functions playing the role of the basic field.  The functions are not central in the differential operators and the de Rham differential is not linear over the functions, which makes such a relativization of Koszul duality highly nontrivial and interesting.

With both 2. and 3., there is a major problem that the duality functors do not preserve acyclicity of complexes.  For example, a nonacyclic DG-algebra is sometimes assigned to an acyclic DG-coalgebra, and a nonacyclic complex of D-modules is assigned to an acyclic DG-module over the de Rham complex.  And for curved DG-structures, the notion of quasi-isomorphism does not even make sense.  The general solution is that one has to introduce an equivalence relation more delicate than quasi-isomorphism, particularly on the coalgebra side of the story.  The relevant references include Hinich's <a href="https://arxiv.org/abs/math/9812034">paper</a>, Lefevre-Hasegawa's <a href="https://arxiv.org/abs/math/0310337">thesis</a> and Keller's <a href="http://people.math.jussieu.fr/~keller/publ/index.html#Talks">exposition</a> of some results contained therein, and my recent <a href="https://arxiv.org/abs/0905.2621">preprint</a>, which is supposed to contain state of the art.  Concerning DG-modules over the de Rham complex, there were earlier approaches by <a href="https://dx.doi.org/10.1007/BFb0086264">Kapranov</a> and <a href="http://www.math.uchicago.edu/~mitya/langlands/hitchin/">Beilinson-Drinfeld</a>.