  In this paper, we will show that solutions of the three-dimensional
non-resistive and non-diffusive MHD-Boussinesq system are globally regular if
the initial data is axisymmetric and the swirl components of the velocity and
the magnetic vorticity are zero. Our main result extends previous ones on the
three-dimensional non-resistive MHD system and non-diffusive Boussinesq system,
and the method used here can also be applied to the magnetic
Rayleigh-B\'{e}nard convection system.
