Studied for more than a century, first in the field of geophysics, flows over rotating discs present a great diversity of complex instability behaviours that are not yet fully understood. While the primary instabilities are now well characterized experimentally, theoretically and numerically, their role in the transition mechanisms to turbulence remains an open question that still challenges the scientific community. This article brings together the main results of the literature related to the instabilities over rotating discs, but also in the connected problem of rotating cavities, and reviews the main scenarios currently assumed to describe the flow breakdown to turbulence. A particular focus is on more recent studies of generic flows in rotating cavities bounded by two coaxial rotating discs, that occur in many industrial systems, the performances of which and their improvement are linked to a better understanding of these mechanisms. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)’.