  We review a recently proposed definition of complexity of the structure of
self--gravitating fluids \cite{ch1}, and the criterium to define the simplest
mode of their evolution. We analyze the origin of these concepts and their
possible applications in the study of gravitation collapse. We start by
considering the static spherically symmetric case, extending next the study to
static axially symmetric case. Afterward we consider the non--static
spherically symmetric case. Two possible modes of evolution are proposed to be
the simplest one. One is the homologous conditio,, however, as was shown later
on, it may be useful to relax this last condition to enlarge the set of
possible solutions, by adopting the so-called quasi-homologous condition. As
another example of symmetry, we consider fluids endowed with hyperbolical
symmetry. Exact solutions for static fluid distributions satisfying the
condition of minimal complexity are presented.. An extension of the complexity
factor to the vacuum solutions of the Einstein equations represented by the
Bondi metric is discussed. A complexity hierarchy is established in this case,
ranging from the Minkowski spacetime (the simplest one) to gravitationally
radiating systems (the most complex). Finally we propose a list of questions
which, we believe, deserve to be treated in the future
