The Lambda Cold Dark Matter (LCDM) paradigm has been spectacularly successful at reproducing observations of the cosmic microwave background and of the large scale structure of the Universe. On these scales, the structures contrasted with the theory are in the linear or mildly non-linear regime, where observations are well established and theoretical predictions are robust. LCDM also makes specific predictions in the non-linear regime; in particular, for the evolution, abundance, structure, and substructure of dark matter halos, the sites of galaxy formation. On these non-linear scales a number of potential challenges to LCDM have been identified when confronting the observed internal structure of galaxies with LCDM expectations. I plan to review briefly the status of these challenges and to discuss whether they signal a potential breakdown of the LCDM paradigm or just reflect our incomplete understanding of the complex process of galaxy formation.
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