  This paper is based on a course given by the author at the University of Rome
``La Sapienza'' in the Academic year 2000/2001. The intended aim of the course
was to rapidly introduce, although not in an exhaustive way, the non-expert PhD
student to deformations of compact complex manifolds, from the very beginning
to some recent (i.e. at that time not yet published) results. The goal of these
lectures is to give a soft introduction to extended deformation theory. In view
of the aim (and the hope) of keeping this paper selfcontained, user friendly
and with a tolerating number of pages, we consider only deformations of compact
complex manifolds. Anyhow, most part of the formalism and of the results that
we prove here will apply to many other deformation problems.
