In most astrophysical disks, angular momentum transport and accretion is thought to be mediated by MHD turbulence driven by the magnetorotational instability (MRI). Recent numerical studies of the MRI will be described, including the effect of finite dissipation on the saturation amplitude, comparison of local simulations using the shearing box in very large domains with fully global models, and measurement of the anomalous transport rate of magnetic flux due to turbulence driven by the MRI. The structure and evolution of very diffuse non-radiative accretion flows around compact objects can be strongly affected by kinetic MHD effects such as anisotropic heat conduction and viscosity. The importance of these effects will be demonstrated with the example of convection in thermally stratified atmospheres, where anisotropic heat conduction can drive the magneto-thermal instability (MTI), which could be important in neutron star atmospheres, and hot gas in galaxy clusters. Results from a recent study of the MTI in spherical Bondi accretion flows will be given. These applications use a new, higher-order Godunov method for astrophysical MHD which preserves the divergence-free constraint exactly, conserves total energy, and works with nested and adaptive meshes; this new method will be described briefly.