  This paper deals with the derivation and analysis of the the Hall
Magneto-Hydrodynamic equations. We first provide a derivation of this system
from a two-fluids Euler-Maxwell system for electrons and ions, through a set of
scaling limits. We also propose a kinetic formulation for the Hall-MHD
equations which contains as fluid closure different variants of the Hall-MHD
model. Then, we prove the existence of global weak solutions for the
incompressible viscous resistive Hall-MHD model. We use the particular
structure of the Hall term which has zero contribution to the energy identity.
Finally, we discuss particular solutions in the form of axisymmetric purely
swirling magnetic fields and propose some regularization of the Hall equation.
