  Though usually treated in isolation, the magnetorotational and gravitational
instabilities (MRI and GI) may coincide at certain radii and evolutionary
stages of protoplanetary discs and active galactic nuclei. Their mutual
interactions could profoundly influence several important processes, such as
accretion variability and outbursts, fragmentation and disc truncation, or
large-scale magnetic field production. Direct numerical simulations of both
instabilities are computationally challenging and remain relatively unexplored.
In this paper, we aim to redress this neglect via a set of 3D vertically
stratified shearing-box simulations, combining self-gravity and magnetic
fields. We show that gravito-turbulence greatly weakens the zero-net-flux MRI.
In the limit of efficient cooling (and thus enhanced GI), the MRI is completely
suppressed, and yet strong magnetic fields are sustained by the
gravitoturbulence. This turbulent `spiral wave' dynamo may have widespread
application, especially in galactic discs. Finally, we present preliminary work
showing that a strong net-vertical-flux revives the MRI and supports a
magnetically dominated state, in which the GI is secondary.
