  Turbulence is the most common state of astrophysical flows. In typical
astrophysical fluids, turbulence is accompanied by strong magnetic fields,
which has a large impact on the dynamics of the turbulent cascade. Recently,
there has been a significant breakthrough on the theory of magnetohydrodynamic
(MHD) turbulence. For the first time we have a scaling model that is supported
by both observations and numerical simulations. We review recent progress in
studies of both incompressible and compressible turbulence. We compare
Iroshnikov-Kraichnan and Goldreich-Sridhar models, and discuss scalings of
Alfv\'en, slow, and fast waves. We also discuss the completely new regime of
MHD turbulence that happens below the scale at which hydrodynamic turbulent
motions are damped by viscosity. In the case of the partially ionized diffuse
interstellar gas the viscosity is due to neutrals and truncates the turbulent
cascade at $\sim$parsec scales. We show that below this scale magnetic
fluctuations with a shallow spectrum persist and discuss the possibility of a
resumption of the MHD cascade after ions and neutrals decouple. We discuss the
implications of this new insight into MHD turbulence for cosmic ray transport,
grain dynamics, etc., and how to test theoretical predictions against
observations.
