  Certain classes of astrophysical objects, namely magnetars and central
engines of supernovae and gamma-ray bursts (GRBs), are characterized by extreme
physical conditions not encountered elsewhere in the Universe. In particular,
they possess magnetic fields that exceed the critical quantum field of 44
teragauss. Figuring out how these complex ultra-magnetized systems work
requires understanding various plasma processes, both small-scale kinetic and
large-scale magnetohydrodynamic (MHD). However, an ultra-strong magnetic field
modifies the underlying physics to such an extent that many relevant
plasma-physical problems call for building QED-based relativistic quantum
plasma physics. In this review, after describing the extreme astrophysical
systems of interest and identifying the key relevant plasma-physical problems,
we survey the recent progress in the development of such a theory. We discuss
how a super-critical field modifies the properties of vacuum and matter and
outline the basic theoretical framework for describing both non-relativistic
and relativistic quantum plasmas. We then turn to astrophysical applications of
relativistic QED plasma physics relevant to magnetar magnetospheres and central
engines of supernovae and long GRBs. Specifically, we discuss propagation of
light through a magnetar magnetosphere; large-scale MHD processes driving
magnetar activity and GRB jet launching and propagation; energy-transport
processes governing the thermodynamics of extreme plasma environments;
micro-scale kinetic plasma processes important in the interaction of intense
magnetospheric electric currents with a magnetar's surface; and magnetic
reconnection of ultra-strong magnetic fields. Finally, we point out that future
progress will require the development of numerical modeling capabilities.
