It turns out that the first divergences in the quantization of general relativity which cannot be absorbed in counterterms consistently (i.e. without the necessity of introducing new parameters) appear already at one-loop level in the presence of matter fields. At two-loop level the problematic divergences arise even in pure gravity. In order to overcome this conceptual difficulty the development of nonperturbative techniques was required, providing various candidate theories of quantum gravity. For a long time the prevailing view has been that the very concept of quantum field theory – even though remarkably successful in the case of the other fundamental interactions – is doomed to failure for gravity. By way of contrast, the idea of asymptotic safety retains quantum fields as the theoretical arena and instead abandons only the traditional program of perturbative renormalization.