The NANOGrav pulsar timing collaboration has recently reported strong evidence for a new stochastic common-spectrum process affecting the pulsar timing residuals in its 12.5-year data set. If confirmed in the future, this signal may turn out to be the first glimpse of a stochastic gravitational-wave background at nanohertz frequencies. In the first half of this talk, I will review the NANOGrav experiment, discuss the properties of the observed signal, and comment on its astrophysical interpretation in terms of inspiraling supermassive black-hole binaries. In the second half of the talk, I will then turn to possible explanations based on physics beyond the standard model, in particular, the possibility of gravitational waves emitted by a network of cosmic strings. In this case, the NANOGRAV signal may be a first peek at the dynamics of the early Universe at energies close to the scale of grand unification. This talk is based on 2009.06607 and 2009.10649.