  In the last years there has been a growing interest in the understanding a
vast variety of scale invariant and critical phenomena occurring in nature.
Experiments and observations indeed suggest that many physical systems develop
spontaneously correlations with power law behaviour both in space and time.
Pattern formation, aggregation phenomena, biological systems, geological
systems, disordered materials, clustering of matter in the universe are just
some of the fields in which scale invariance has been observed as a common and
basic feature. However, the fact that certain structures exhibit fractal and
complex properties does not tell us why this happens. A crucial point to
understand is therefore the origin of the general scale-invariance of natural
phenomena. This would correspond to the understanding of the origin of fractal
structures and of the properties of Self- Organized Criticality (SOC) from the
knowledge of the microscopic physical processes at the basis of these
phenomena. Fractal geometry can play a crucial role in extracting the correct
physical properties from experimental data. In particular, the recent
availability of complete three dimensional samples of galaxies and clusters
permits a direct study of their spatial properties. We present a brief review
of galaxy correlations based on the methods of modern Statistical Physics.
These methods are able to identify self-similar and non-analytical properties,
and allow us to test the usual homogeneity assumption of luminous matter
distribution.
  The new analysis shows that all the available data are consistent with each
other and show fractal correlations (with dimension $D \simeq 2$) up to the
deepest scales probed until now ($1000 \hmp$)
