References for the moduli space of complex structures I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex associated and, more importantly, the differential graded Lie algebra structure that apparently appears in this problem.
I am interested in learning about this machinery in general more than only for the specific case of complex structures, so references about the general set-up are also welcome. In particular, my ultimate goal is to understand how $L_{\infty}$-algebras are associated to deformation problems, and since differential graded Lie algebras are particular instances of $L_{\infty}$-algebras that looks like a good place where to start.
Thanks.
 A: Concerning the deformation theory of complex manifolds, there are of course the seminal papers of Kodaira-Spencer. There are also some more recent notes of Manetti, Lectures on deformations of complex manifolds, which are available on arxiv and could be of interest for you.
The general principle relating deformation problems to DGLAs or more general $L_{\infty}$-algebras has a very long history. The notion of deformation functor goes back to Schlessinger's paper Functors of Artin rings. The general principle that every deformation problem should arise as a deformation functor associated to a certain dgla emerged from unpublished work of Deligne and Drinfeld and found its first striking application in Goldman-Millson's paper The deformation theory of representations of fundamental groups of compact Kähler manifolds.
Briefly, to every dgla $g$ one can associate its set of Maurer-Cartan elements $MC(g)$, on which acts the so called gauge group exp(g^0) (here we have to add a pronilpotency condition on $g$ to have a well defined gauge group). The quotient is called the Maurer-Cartan moduli set, and can be made into a functor from artinian rings to sets which is the deformation functor associated to $g$. This is well explained in Manetti's Deformation theory via differential graded Lie algebras for instance (but there are certainly plenty of other references). Maurer-Cartan elements are the structures you want to study and the gauge group gives you an equivalence relation between such structures.
It turns out that the Maurer-Cartan moduli set is invariant under quasi-isomorphisms of dlgas and more generally under $L_{\infty}$-quasi-isomorphisms, a propery which was crucial in Kontsevich's work Deformation quantization of Poisson manifolds.
Now let us go in a ""derived" world : one can define a simplicial Maurer-Cartan set whose 1-simplices gives a notion of homotopy between two Maurer-Cartan elements, which is equivalent to the action of the gauge group. The main advantage is that such homotopies can be defined for $L_{\infty}$-algebras, for which there is no gauge groupe anymore.
The simplicial Maurer-Cartan set construction and several properties of it are well explained in Getzler's Lie theory for nilpotent $L_{\infty}$-algebras.
Like the Maurer-Cartan moduli set, it is invariant (up to homotopy) under quasi-isomorphisms of $L_{\infty}$-algebras, and even under $L_{\infty}$-quasi-isomorphisms according to a recent paper of Dolgushev-Rogers A Version of the Goldman-Millson Theorem for Filtered L-infinity Algebras.
The "Deligne principle" of deformation theory was stated rigorously as a theorem only recently (several years ago) in the realm of derived algebraic geometry, as an equivalence of $\infty$-categories between dglas and moduli problems. This is due to Lurie in Derived algebraic geometry X and Pridham in Unifying derived deformation theories.
A: For the underlying DGLA, take a look at The homotopy invariance of the Kuranishi space by Goldman and Millson, and also an earlier paper by the same authors in IHES.
You can of course go back to the original papers of Kodaira-Spencer and Kuranishi for  general background.
