  Accretion discs are present around both stellar-mass black holes in X-ray
binaries and supermassive black holes in active galactic nuclei. A wide variety
of circumstantial evidence implies that many of these discs are warped. The
standard Bardeen--Petterson model attributes the shape of the warp to the
competition between Lense--Thirring torque from the central black hole and
viscous angular-momentum transport within the disc. We show that this
description is incomplete, and that torques from the companion star (for X-ray
binaries) or the self-gravity of the disc (for active galactic nuclei) can play
a major role in determining the properties of the warped disc. Including these
effects leads to a rich set of new phenomena. For example, (i) when a companion
star is present and the warp arises from a misalignment between the companion's
orbital axis and the black hole's spin axis, there is no steady-state solution
of the Pringle--Ogilvie equations for a thin warped disc when the viscosity
falls below a critical value; (ii) in AGN accretion discs, the warp can excite
short-wavelength bending waves that propagate inward with growing amplitude
until they are damped by the disc viscosity. We show that both phenomena can
occur for plausible values of the black hole and disc parameters, and briefly
discuss their observational implications.
