  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.
