Recent results from simulations of the magnetorotational instability

The nonlinear saturation of the magnetorotational instability (MRI) is best studied through numerical MHD simulations. Recent results of simulations that adopt the local shearing box approximation, and fully global models that follow the entire disk, are described. Outstanding issues remain, such as a first-principles understanding of the dynamo processes that control saturation with no net magnetic flux. Important directions for future work include a better understanding of basic plasma processes, such as reconnection, dissipation, and particle acceleration, in the MHD turbulence driven by the MRI.