I am reading about deformation theory. I am treating mostly the algebraic case, but I would like to know a bit about all facets of this field of mathematics, so the geometric case is also of great interest to me. What are good references for the theory of deformations of complex analytic structures on a manifold?
Remark: I have not yet read the original papers (Kodaira-Spencer, Frölicher-Nijenhuis, if I remember well, or some other combinations of these four names...), but maybe I'd like to start from a more textbook-like reference, and not from research articles.
A nice introductory reference for the theory of deformation of complex structures is chapter $6$ of Huybrechts' Complex Geometry: An Introduction.
In section $1$, deformations are studied as smoothly varying almost complex structures on a fixed smooth manifold subject to integrability which leads to the Maurer-Cartan equation. I asked for further references for this approach in this MathOverflow question.
In section $2$, a deformation is a smooth proper holomorphic map between connected complex manifolds - by Ehresmann's Theorem, this approach coincides with the one mentioned in the previous paragraph.
What isn't covered in this book is the coordinate approach developed by Kodaira and Spencer.