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 identify the 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. In particular we discuss the properties of the following catalogs: CfA, Perseus-Pisces, SSRS, IRAS, LEDA, APM-Stromlo, Las Campanas and ESP for galaxies and Abell and ACO for galaxy clusters. 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 similar or equal to 2) up to the deepest scales probed so far (1000 h(-1) Mpc) 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 h(-1) Mpc due to the statistical robustness of the data but becomes progressively weaker (statistically) at larger distances due to the limited data. In addition, the luminosity distribution is correlated with the space distribution in a specific way. 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. (C) 1998 Elsevier Science B.V.
