It is well known that homotopical localizations with respect to sets of maps exist in left proper combinatorial model categories, and the same result holds for classes of maps assuming the validity of Vopenka's principle. In a joint work with Javier Gutierrez, we prove that the existence of localization with respect to every class of maps in the homotopy category of simplicial sets or in the homotopy category of spectra is equivalent to the weak Vopenka principle.