We discuss results of global three-dimensional magnetohydrodynamic simulations of accretion on to a rotating magnetized star with a tilted dipole magnetic field, where the accretion is driven by the magnetorotational instability (MRI). The simulations show that MRI-driven turbulence develops in the disc, and angular momentum is transported outwards primarily due 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 magnetic field of the star is dynamically important: the accretion disc is truncated by the star’s magnetic field within a few stellar radii from the star’s surface, and matter flows to the star in funnel streams. The funnel streams flow towards the south and north magnetic poles but are not equal due to the inhomogeneity of the flow. The hotspots on the stellar surface are not symmetric as well. In the BL regime, the magnetic field of the star is dynamically unimportant, and matter accretes on to the surface of the star through the BL. 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 hot region on the surface of the star. The belt has inhomogeneous density distribution which varies in time due to variable accretion rate. The peaks in the variability curve are associated with accretion of individual turbulent cells. They show 20–50 per cent density amplifications at periods of ∼5–10 dynamical time-scales at the surface of the star. Spiral waves in the disc are excited in both magnetospheric and BL regimes of accretion. 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.