  The $\Lambda$CDM 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. $\Lambda$CDM 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 $\Lambda$DCM 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 $\Lambda$CDM 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.
