Abstract We review the works devoted to third and fifth harmonic susceptibilities in glasses, namely and . We explain why these nonlinear responses are especially well adapted to test whether or not some amorphous correlations develop upon cooling. We show that the experimental frequency and temperature dependences of and of have anomalous features, since their behavior is qualitatively different to that of an ideal dipolar gas, which is the high temperature limit of a fluid. Most of the works have interpreted this anomalous behavior as reflecting the growth, upon cooling, of amorphously ordered domains, as predicted by the general framework of Bouchaud and Biroli (BB). We explain why most—if not all—of the challenging interpretations can be recast in a way which is consistent with that of BB. Finally, the comparison of the anomalous features of and of shows that the amorphously ordered domains are compact, i.e. the fractal dimension d f is close to the dimension d of space. This suggests that the glass transition of molecular liquids corresponds to a new universality class of critical phenomena.