  A concise review is given of astrophysically motivated experimental and
theoretical research on Taylor-Couette flow. The flows of interest rotate
differentially with inner cylinder faster than outer one but are linearly
stable against Rayleigh's inviscid centrifugal instability. At shear Reynolds
numbers as large as 10^6, hydrodynamic flows of this type (quasi-keplerian)
appear to be nonlinearly stable: no turbulence is seen that cannot be
attributed to interaction with the axial boundaries, rather than the radial
shear itself. Direct numerical simulations agree, although they cannot yet
reach such high Reynolds numbers. This result indicates that accretion-disc
turbulence is not purely hydrodynamic in origin, at least insofar as it is
driven by radial shear. Theory, however, predicts linear magnetohydrodynamic
(MHD) instabilities in astrophysical discs: in particular, the standard
magnetorotational instability (SMRI). MHD Taylor-Couette experiments aimed at
SMRI are challenged by the low magnetic Prandtl numbers of liquid metals. High
fluid Reynolds numbers and careful control of the axial boundaries are
required. The quest for laboratory SMRI has been rewarded with the discovery of
some interesting inductionless cousins of SMRI, and the recently reported
success in demonstrating SMRI itself by taking advantage of conducting axial
boundaries. Some outstanding questions and near-future prospects are discussed,
especially in connection with astrophysics.
