Studying the physical processes occurring in the region just above the magnetic poles of strongly magnetized, accreting binary neutron stars is essential to our understanding of stellar and binary system evolution. Perhaps more importantly, it provides us with a natural laboratory for studying the physics of high temperature and high density plasmas exposed to extreme radiation, gravitational, and magnetic fields. Observations over the past decade have shed new light on the manner in which plasma falling at velocities near the speed of light onto a neutron star surface is halted. Recent advances in modeling these processes have resulted in direct measurement of the magnetic fields and plasma properties. On the other hand, numerous physical processes have been identified that challenge our current picture of how the accretion process onto neutron stars works. Observation and theory are our essential tools in this regime because the extreme conditions cannot be duplicated on Earth. This white paper gives an overview of the current theory, the outstanding theoretical and observational challenges, and the importance of addressing them in contemporary astrophysics research.
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