  The study of K\"ahler immersions of a given real analytic K\"ahler manifold
into a finite or infinite dimensional complex space form originates from the
pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines
a powerful tool, a special (local) potential called diastasis function, which
allows him to obtain necessary and sufficient conditions for a neighbourhood of
a point to be locally K\"ahler immersed into a finite or infinite dimensional
complex space form. As application of its criterion, he also provides a
classification of (finite dimensional) complex space forms admitting a K\"ahler
immersion into another. Although, a complete classification of K\"ahler
manifolds admitting a K\"ahler immersion into complex space forms is not known,
not even when the K\"ahler manifolds involved are of great interest, e.g. when
they are K\"ahler-Einstein or homogeneous spaces. In fact, the diastasis
function is not always explicitely given and Calabi's criterion, although
theoretically impeccable, most of the time is of difficult application.
Nevertheless, throughout the last 60 years many mathematicians have worked on
the subject and many interesting results have been obtained.
  The aim of this book is to describe Calabi's original work, to provide a
detailed account of what is known today on the subject and to point out some
open problems.
