The magnetohydrodynamic (MHD) equations model electrically conducting fluids such as plasmas and are important in understanding many natural phenomena such as solar flares and the formation of Northern Lights. Mathematically the MHD equations can be difficult to analyze due to the nonlinear coupling between the induction equation and the Navier-Stokes equations with the Lorentz force. One fundamental problem on the MHD equations is whether or not their solutions exist for all time. This problem has attracted considerable interest recently for the 2D MHD equations with partial dissipation. This talk presents some very recent global regularity results for various partial dissipation cases.