The observation of large-amplitude ‘kink’ waves on the vortex cores produced by an oscillating grid in a rotating fluid (Hopfinger, Browand & Gagne 1982) has motivated the study of such waves under more controlled circumstances. We have experimentally observed the properties of helical waves, rotating, plane standing waves and evolving, isolated kink-waves. Their characteristics have been related to theories based on the localized induction equation of Arms & Hama (1965), the ‘cut-off’ theory of Crow (1970) as extended by Moore & Saffman (1972), and an extension of Pocklington's (1895) dispersion relationship for ‘hollow-core’ vortices. It is shown that the latter dispersion relation and the Moore & Saffman theory are good approximations to our experimental results. Using these, we present new results on solitary kink-wave properties of concentrated vortex flows, and in particular show that envelope solitons are possible only for a restricted range of carrier wavenumbers. A second class of waves was also observed: the axisymmetric solitary waves of Benjamin (1967). These were found to become unstable to spiral disturbances when their amplitude exceeded a certain magnitude, as has been found in the study of the related vortex-breakdown phenomenon. All of these observations are used to interpret the experiments presented by HBG and to discuss qualitatively the dynamics of rotating turbulence. In the Appendix we propose a possible mechanism by which concentrated vortices can be formed in a rotating turbulent fluid.