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that every deformation problem should arise as a deformation functor associated to a certain dgla emerged from unpublished work of Deligne and Drinfeld and found its first striking application in Goldman-Millson's paper *The deformation theory of representations of fundamental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).