Pulsar timing is a sensitive tool of modern astrophysical research enabling measurements of the time delay of electromagnetic signals propagating from the source to the observer. Modern pulsar timing arrays (PTAs) are used to address various astrophysical problems, including the direct detection of space—time metric perturbations caused, in particular, by gravitational waves. We review the current status of pulsar timing research and recent results on the detection of a stochastic gravitational wave background at nanohertz frequencies of astrophysical and cosmological origin reported by the international collaborations NANOGrav, EPTA, InPTA, PPTA, and CPTA. We also discuss current constraints on scalar ultralight dark matter (pseudoscalar axion-like bosons) posed by the pulsar timing and polarimetry and prospects of these methods to solve other problems of fundamental physics and cosmology.