  With the fundamental stress mechanism of accretion disks
identified--correlated MHD turbulence driven by the magneto-rotational
instability--it has become possible to make numerical simulations of accretion
disk dynamics based on well-understood physics. A sampling of results from both
Newtonian 3-d shearing box and general relativistic global disk MHD simulations
is reported. Among other things, these simulations have shown that: contrary to
long-held assumptions, stress is continuous through the marginally stable and
plunging regions around black holes, so that rotating black holes can give
substantial amounts of angular momentum electromagnetically to surrounding
matter; the upper layers of accretion disks are primarily supported by magnetic
pressure, potentially leading to interesting departures from local black-body
emitted spectra; and initially local magnetic fields in accretion flows can, in
some cases, spontaneously generate large-scale fields that connect rotating
black holes to infinity and mediate strong relativistic jets.
