Pulsating ULXs: highly magnetised neutron stars at extreme mass accretion rates

X-ray pulsars (XRPs) form a special class in a family of accreting neutron stars (NSs). They stand out from the other classes due to their strong magnetic field, which typically exceeds 10^{12} G and affects even fundamental properties of matter. Magnetic field funnels the accretion flow and its gravitational energy is released in the form of X-rays coming from the compact area on the NS surface. Recent discoveries of pulsations from ultra-luminous X-ray sources (ULXs) have opened a new chapter in studies of XRPs. The classical theoretical limitation for luminosity is given by the Eddington value, which is about 2× 10^{38} erg/s for NSs. Discovery of ULXs powered by accreting NSs is a challenge for theoretical astrophysics opening new questions on how accreting NSs can exceed the Eddington limit by at least two orders of magnitude. I will discuss the features of XRPs, which arise and become essential at extremely high mass accretion rates: (1) accretion columns, which arise at super-critical mass accretion rates and provide a principal possibility to exceed the Eddington value, (2) optically thick envelopes, which are formed by hot accretion flow at the magnetospheric surface and can affect the spectral and timing properties of ULX pulsars, and (3) possibly strong neutrino emission at extreme mass accretion rates.