  Shell models of hydrodynamic turbulence originated in the seventies. Their
main aim was to describe the statistics of homogeneous and isotropic turbulence
in spectral space, using a simple set of ordinary differential equations. In
the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged
based on the same principles as their hydrodynamic counter-part but also
incorporating interactions between magnetic and velocity fields. In recent
years, significant improvements have been made such as the inclusion of
non-local interactions and appropriate definitions for helicities. Though shell
models cannot account for the spatial complexity of MHD turbulence, their
dynamics are not over simplified and do reflect those of real MHD turbulence
including intermittency or chaotic reversals of large-scale modes. Furthermore,
these models use realistic values for dimensionless parameters (high kinetic
and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing
extended inertial range and accurate dissipation rate. Using modern computers
it is difficult to attain an inertial range of three decades with direct
numerical simulations, whereas eight are possible using shell models. In this
review we set up a general mathematical framework allowing the description of
any MHD shell model. The variety of the latter, with their advantages and
weaknesses, is introduced. Finally we consider a number of applications,
dealing with free-decaying MHD turbulence, dynamo action, Alfven waves and the
Hall effect.
