  Some years ago we proposed a new approach to the analysis of galaxy and
cluster correlations based on the concepts and methods of modern statistical
Physics. This led to the surprising result that galaxy correlations are fractal
and not homogeneous up to the limits of the available catalogs. The usual
statistical methods, which are based on the assumption of homogeneity, are
therefore inconsistent for all the length scales probed so far, and a new, more
general, conceptual framework is necessary to identifythe real physical
properties of these structures. In the last few years the 3-d catalogs have
been significatively improved and we have extended our methods to the analysis
of number counts and angular catalogs. This has led to a complete analysis of
all the available data that we present in this review. The result is that
galaxy structures are highly irregular and self-similar: all the available data
are consistent with each other and show fractal correlations (with dimension $D
\simeq 2$) up to the deepest scales probed so far ($1000 \hmp$) and even more
as indicated from the new interpretation of the number counts. The evidence for
scale-invariance of galaxy clustering is very strong up to $150 \hmp$ due to
the statistical robustness of the data but becomes progressively weaker
(statistically) at larger distances due to the limited data. In These facts
lead to fascinating conceptual implications about our knowledge of the universe
and to a new scenario for the theoretical challenge in this field.
