  Fractal structures are observed in the universe in two very different ways.
Firstly, in the gas forming the cold interstellar medium in scales from 10^{-4}
pc till 100 pc. Secondly, the galaxy distribution has been observed to be
fractal in scales up to hundreds of Mpc. We give here a short review of the
statistical mechanical (and field theoretical) approach developed by us. We
consider a non-relativistic self-gravitating gas in thermal equilibrium at
temperature T inside a volume V. The statistical mechanics of such system has
special features and, as is known, the thermodynamical limit does not exist in
its customary form. Moreover, the treatments through microcanonical, canonical
and grand canonical ensembles yield different results.We present here for the
first time the equation of state for the self-gravitating gas in the canonical
ensemble. We find that it has the form p = [N T/ V] f(eta), where p is the
pressure, N is the number of particles and \eta \equiv {G m^2 N \over V^{1/3}
T}. The N \to\infty and V \to\infty limit exists keeping \eta fixed. We compute
the function f(\eta) using Monte Carlo simulations and for small eta
analytically. We compute the thermodynamic quantities of the system as free
energy, entropy, chemical potential, specific heat, compressibility and speed
of sound. We reproduce the well-known gravitational phase transition associated
to the Jeans' instability. Namely, a gaseous phase for eta < eta_c and a
condensed phase for eta > eta_c. Moreover, we derive the precise behaviour of
the physical quantities near the transition. In particular, the pressure
vanishes as p \sim(eta_c-eta)^B with B \sim 0.2 and eta_c \sim 1.6 and the
energy fluctuations diverge as \sim(eta_c-eta)^{B-1}. The speed of sound
decreases monotonically and approaches the value sqrt{T/6} at the transition.
