X-ray pulsars (XRPs) are accreting strongly magnetized 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 systems is the magnetic field strength at their surface, about or even higher than 10¹² 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.