Resulting from the availability of improved sources, research in the terahertz (THz) spectral range has increased dramatically over the last decade, leading essentially to the disappearance of the so-called 'THz gap'. While most work to date has been carried out with THz radiation of low field amplitude, a growing number of experiments are using THz radiation with large electric and magnetic fields that induce nonlinearities in the system under study. This 'focus on' collection contains a number of articles, both experimental and theoretical, in the new subfield of THz nonlinear optics and spectroscopy on various systems, among them molecular gases, superconductors, semiconductors, antiferromagnets and graphene.A nonlinear response to the electromagnetic field requires high intensity. At first glance, it may seem improbable to achieve sufficiently large intensities (or electric field amplitudes) in the THz