  X-ray pulsars (XRPs) are accreting strongly magnetised neutron stars (NSs) in
binary systems with, as a rule, massive optical companions. Very reach
phenomenology and high observed flux put them into the focus of observational
and theoretical studies since the first X-ray instruments were launched into
space. The main attracting characteristic of NSs in this kind of system is the
magnetic field strength at their surface, about or even higher than
$10^{12}\,{\rm G}$, that is about six orders of magnitude stronger than what is
attainable in terrestrial laboratories. Although accreting XRPs were discovered
about 50 years ago, the details of the physical mechanisms responsible for
their properties are still under debate. Here we review recent progress in
observational and theoretical investigations of XRPs as a unique laboratory for
studies of fundamental physics (plasma physics, QED and radiative processes)
under extreme conditions of ultra-strong magnetic field, high temperature, and
enormous mass density.
