Every finite-volume hyperbolic manifold is finitely covered by a stably parallelizable manifold, which in particular has trivial Stiefel-Whitney and Pontryagin classes. On the other hand, it is difficult to produce hyperbolic manifolds with non-trivial characteristic classes. In two recent works, one with Rizzi and one with Bustamante and Reyes, we complement the previously existing results of Long-Reid, Martelli-R-Slavich and Chen on the theme, solving one of the K3 problems and answering questions of Charney-Davis and of Belegradek on strict hyperbolization, respectively. The talk will essentially consist of an overview on the subject.