  Accreting neutron stars (NSs) represent a unique laboratory for probing the
physics of accretion in the presence of strong magnetic fields ($B\gtrsim 10^8$
G). Additionally, the matter inside the NS itself exists in an ultra-dense,
cold state that cannot be reproduced in Earth-based laboratories. Hence,
observational studies of these objects are a way to probe the most extreme
physical regimes. Here we present an overview of the field and discuss the most
important outstanding problems related to NS accretion. We show how these open
questions regarding accreting NSs in both low-mass and high-mass X-ray binary
systems can be addressed with the High-Energy X-ray Probe (HEX-P) via simulated
data. In particular, with the broad X-ray passband and improved sensitivity
afforded by a low X-ray background, HEX-P will be able to 1) distinguish
between competing continuum emission models; 2) provide tighter upper limits on
NS radii via reflection modeling techniques that are independent and
complementary to other existing methods; 3) constrain magnetic field geometry,
plasma parameters, and accretion column emission patterns by characterizing
fundamental and harmonic cyclotron lines and exploring their behavior with
pulse phase; 4) directly measure the surface magnetic field strength of highly
magnetized NSs at the lowest accretion luminosities; as well as 5) detect
cyclotron line features in extragalactic sources and probe their dependence on
luminosity in the super-Eddington regime in order to distinguish between
geometrical evolution and accretion-induced decay of the magnetic field. In
these ways HEX-P will provide an essential new tool for exploring the physics
of NSs, their magnetic fields, and the physics of extreme accretion.
