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The recent observations of galaxy and dark matter clumpy distributions have provided new elements to the understating of the problem of cosmological structure formation. The strong clumpiness characterizing galaxy structures seems to be present in the overall mass distribution and its relation to the highly isotropic Cosmic Microwave Background Radiation represents a fundamental problem. The extension of structures, the formation of power-law correlations characterizing the strongly clustered regime and the relation between dark and visible matter are the key problems both from an observational and a theoretical point of view. We discuss recent progresses in the studies of structure formation by using concepts and methods of statistical physics.

Ideas of Statistical Physics are very relevant for cosmic structures especially considering that the field is undergoing a period of exceptional development with many new data appearing on a monthly basis. In the past years we have focused mostly on galaxy distributions and their statistical properties. This led to an interesting debate which will be resolved by the next generation of data in a couple of years. In addition here we discuss the statistical properties of the fluctuations of the cosmic microwave background which are small in amplitude but complex in structure. We finally discuss the connection between these observations and the Harrison-Zeldovich spectrum and its further implications on the theories of structure formation and the and the cosmological $N$-body simulations.

Cosmic structures at large scales represent the earliest and most extended form of matter condensation. In this lecture we review the application of the methods and concepts of modern statistical physics to these structures. This leads to a new perspective in the field which can be tested by the many new data which are appearing in the near future. In particular, galaxy structures show fractal correlation up to the present observational limits. The cosmic microwave background radiation, which should trace the initial conditions from which these structures have emerged through gravitational dynamics, is instead extremely smooth. Understanding the relation between the complex galaxy structures and the smooth microwave background radiation represents an extremely challenging problem in the field of structure formation.