  Theoretical approaches to low-frequency magnetized turbulence in
collisionless and weakly collisional astrophysical plasmas are reviewed. The
proper starting point for an analytical description of these plasmas is kinetic
theory, not fluid equations. The anisotropy of the turbulence is used to
systematically derive a series of reduced analytical models. Above the ion
gyroscale, it is shown rigourously that the Alfven waves decouple from the
electron-density and magnetic-field-strength fluctuations and satisfy the
Reduced MHD equations. The density and field-strength fluctuations (slow waves
and the entropy mode in the fluid limit), determined kinetically, are passively
mixed by the Alfven waves. The resulting hybrid fluid-kinetic description of
the low-frequency turbulence is valid independently of collisionality. Below
the ion gyroscale, the turbulent cascade is partially converted into a cascade
of kinetic Alfven waves, damped at the electron gyroscale. This cascade is
described by a pair of fluid-like equations, which are a reduced version of the
Electron MHD. The development of these theoretical models is motivated by
observations of the turbulence in the solar wind and interstellar medium. In
the latter case, the turbulence is spatially inhomogeneous and the anisotropic
Alfvenic turbulence in the presence of a strong mean field may coexist with
isotropic MHD turbulence that has no mean field.
