  We present a theoretical framework for plasma turbulence in astrophysical
plasmas (solar wind, interstellar medium, galaxy clusters, accretion disks).
The key assumptions are that the turbulence is anisotropic with respect to the
mean magnetic field and frequencies are low compared to the ion cyclotron
frequency. The energy injected at the outer scale scale has to be converted
into heat, which ultimately cannot be done without collisions. A KINETIC
CASCADE develops that brings the energy to collisional scales both in space and
velocity. Its nature depends on the physics of plasma fluctuations. In each of
the physically distinct scale ranges, the kinetic problem is systematically
reduced to a more tractable set of equations. In the "inertial range" above the
ion gyroscale, the kinetic cascade splits into a cascade of Alfvenic
fluctuations, which are governed by the RMHD equations at both the collisional
and collisionless scales, and a passive cascade of compressive fluctuations,
which obey a linear kinetic equation along the moving field lines associated
with the Alfvenic component. In the "dissipation range" between the ion and
electron gyroscales, there are again two cascades: the kinetic-Alfven-wave
(KAW) cascade governed by two fluid-like Electron RMHD equations and a passive
phase-space cascade of ion entropy fluctuations. The latter cascade brings the
energy of the inertial-range fluctuations that was damped by collisionless
wave-particle interaction at the ion gyroscale to collisional scales in the
phase space and leads to ion heating. The KAW energy is similarly damped at the
electron gyroscale and converted into electron heat. Kolmogorov-style scaling
relations are derived for these cascades. Astrophysical and space-physical
applications are discussed in detail.
