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 the recent results about the detection of a stochastic gravitational wave background at nanohertz frequencies of astrophysical and cosmological origin reported by international collaborations NANOGrav, EPTA, InPTA, PPTA and CPTA. We also discuss the current constraints on the 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.