Toby Wood Newcastle University Applied Mathematics Neutron stars have the strongest magnetic fields in the universe, but the magneto-hydrodynamics of neutron stars has not been studied until quite recently. The magnetic field is generated by the flow of electrons in the outer "crust" of the star, in which the ions are compressed by gravity into a solid crystalline lattice. We will summarize what is known (and not known) about instabilities and turbulence in this "electron-MHD" regime, and discuss the implications for the rotational and thermal evolution of neutron stars.