  The goal of this minireview is restricted to describe how the
Kolmogorov-Johnson-Mehl-Avrami model has evolved from its birth up to the
present day. The model, which dates back to the late of 1930s, has the purpose
of describing the kinetics of a phase transformation. Given the nature of this
article, although there are hundreds (if not thousands) of experimental data
concerning the most disparate topics, which are interpreted on the basis of the
KJMA model, no arguments relating to these, will be touched upon. Starting from
the ingenious concept of phantom nuclei, firstly introduced by Avrami to get
the exact kinetics, we review theoretical approaches which overcome such
concept. We will show how spatial correlation among nuclei plays a fundamental
role in these modelings.
