  As is well-known, for plasmas of high density and modest temperature, the
classical kinetic theory needs to be extended. Such extensions can be based on
the Schr\"odinger Hamiltonian, applying a Wigner transform of the density
matrix, in which case the Vlasov equation is replaced by the celebrated
Wigner-Moyal equation. Extending the treatment to more complicated models, we
investigate aspects such as spin dynamics (based on the Pauli Hamiltonian),
exchange effects (using the Hartree-Fock approximation), Landau quantization,
and quantum relativistic theory. In the relativistic theory we first study
cases where the field strength is well beyond Schwingers critical field. Both
weakly relativistic theory (gamma factors close to unity) and strongly
relativistic theory are investigated, using assumptions that allow for a
separation of electron and positron states. Finally, we study the
Dirac-Heisenberg-Wigner (DHW) formalism, which is a fully quantum relativistic
theory, allowing for field strengths of the order of the Schwinger critical
field or even larger. As a result, the quantum kinetic theory is extended to
cover phenomena such as Zitterbewegung and electron-positron pair creation.
While the focus of this review is on the quantum kinetic models, we illustrate
the theories with various applications throughout the manuscript.
