The ΛCDM model, or concordance cosmology, as it is often called, is a paradigm at its maturity. It is clearly able to describe the universe at large scale, even if some issues remain open, such as the cosmological constant problem, the small-scale problems in galaxy formation, or the unexplained anomalies in the CMB. ΛCDM clearly shows difficulty at small scales, which could be related to our scant understanding, from the nature of dark matter to that of gravity; or to the role of baryon physics, which is not well understood and implemented in simulation codes or in semi-analytic models. At this stage, it is of fundamental importance to understand whether the problems encountered by the ΛDCM model are a sign of its limits or a sign of our failures in getting the finer details right. In the present paper, we will review the small-scale problems of the ΛCDM model, and we will discuss the proposed solutions and to what extent they are able to give us a theory accurately describing the phenomena in the complete range of scale of the observed universe.
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