  Enormous progress has been made on observing stellar magnetism in stars from
the main sequence through to compact objects. Recent data have thrown into
sharper relief the vexed question of the origin of stellar magnetic fields,
which remains one of the main unanswered questions in astrophysics. In this
chapter we review recent work in this area of research. In particular, we look
at the fossil field hypothesis which links magnetism in compact stars to
magnetism in main sequence and pre-main sequence stars and we consider why its
feasibility has now been questioned particularly in the context of highly
magnetic white dwarfs. We also review the fossil versus dynamo debate in the
context of neutron stars and the roles played by key physical processes such as
buoyancy, helicity, and superfluid turbulence,in the generation and stability
of neutron star fields.
  Independent information on the internal magnetic field of neutron stars will
come from future gravitational wave detections. Thus we maybe at the dawn of a
new era of exciting discoveries in compact star magnetism driven by the opening
of a new, non-electromagnetic observational window.
  We also review recent advances in the theory and computation of
magnetohydrodynamic turbulence as it applies to stellar magnetism and dynamo
theory. These advances offer insight into the action of stellar dynamos as well
as processes whichcontrol the diffusive magnetic flux transport in stars.
