  This work provide a thorough study of L\'evy or heavy-tailed random matrices
(LM). By analysing the self-consistent equation on the probability distribution
of the diagonal elements of the resolvent we establish the equation determining
the localisation transition and obtain the phase diagram of LMs. Using
arguments based on super-symmetric field theory and Dyson Brownian motion we
show that the eigenvalue statistics is the same one of the Gaussian Orthogonal
Ensemble in the whole delocalised phase and is Poisson in the localised phase.
Our numerics confirms these findings, valid in the limit of infinitely large
LMs, but also reveals that the characteristic scale governing finite size
effects diverges much faster than a power law approaching the transition and is
already very large far from it. This leads to a very wide cross-over region in
which the system looks as if it were in a mixed phase. Our results, together
with the ones obtained previously, provide now a complete theory of L\'evy
matrices.
