  We discuss results of global 3D MHD simulations of accretion on to a rotating
magnetized star with a tilted dipole magnetic field, where the accretion is
driven by the magneto-rotational instability (MRI). The simulations show that
MRI-driven turbulence develops in the disc, and angular momentum is transported
outwards due primarily to the magnetic stress. The turbulent flow is strongly
inhomogeneous and the densest matter is in azimuthally-stretched turbulent
cells. We investigate two regimes of accretion: a magnetospheric regime and a
boundary layer (BL) regime. In the magnetospheric regime, the accretion disc is
truncated by the star's magnetic field within a few stellar radii from the
star, and matter flows to the star in funnel streams. The funnel streams
flowing towards the south and north magnetic poles but are not equal due to the
inhomogeneity of the flow. In the BL regime, matter accretes to the surface of
the star through the boundary layer. The magnetic field in the inner disc is
strongly amplified by the shear of the accretion flow, and the matter and
magnetic stresses become comparable. Accreting matter forms a belt-shaped
region on the surface of the star. The belt has inhomogeneous density
distribution which varies in time due to variable accretion rate. Results of
simulations can be applied to classical T Tauri stars, accreting brown dwarfs,
millisecond pulsars, dwarf novae cataclysmic variables, and other stars with
magnetospheres smaller than several stellar radii.
