This talk presents recent work on the global regularity problem
concerning the two-dimensional (2D) magnetohydrodynamic (MHD)
equations with only partial or fractional dissipation. Due to the
nonlinear coupling between the evolution equations of the velocity
and the magnetic field in the MHD system, the global regularity problem
can be extremely difficult when only partial dissipation is present.
We will report recent efforts on several partial or fractional dissipation
cases and will cover both incompressible and compressible flows.