  Taylor-Couette flow -- the flow between two coaxial co- or counter-rotating
cylinders -- is one of the paradigmatic systems in physics of fluids. The
(dimensionless) control parameters are the Reynolds numbers of the inner and
outer cylinder, the ratio of the cylinder radii, and the aspect ratio. The
response of the system is the torque required to retain constant angular
velocities, which can be connected to the angular velocity transport through
the gap. While the low Reynolds number regime has been very well explored in
the '80s and '90s of the last century, in the fully turbulent regime major
research activity only developed in the last decade. In this paper we review
this recent progress in our understanding of fully developed Taylor-Couette
turbulence, from the experimental, numerical, and theoretical point of view. We
will focus on the parameter dependence of the global torque and on the local
flow organisation, including velocity profiles and boundary layers. Next, we
will discuss transitions between different (turbulent) flow states. We will
also elaborate on the relevance of this system for astrophysical disks
(Keplerian flows). The review ends with a list of challenges for future
research on turbulent Taylor-Couette flow.
  The published version in ARFM: Annu. Rev. Fluid Mech. 48:53-80 (2016).
