We have used the technique of laser-Doppler velocimetry to study the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. The experiment was designed to test recent proposals for the number and types of dynamical regimes exhibited by a flow before it becomes turbulent. For different Reynolds numbers the radial component of the local velocity was recorded as a function of time in a computer, and the records were then Fourier-transformed to obtain velocity power spectra. The first two instabilities in the flow, to time-independent Taylor vortex flow and then to time-dependent wavy vortex flow, are well known, but the present experiment provides the first quantitative information on the subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex flow the velocity spectra contain a single sharp frequency component and its harmonics; the flow is strictly periodic. As the Reynolds number is increased, a previously unobserved second sharp frequency component appears at R/Rc = 10·1, where Rc is the critical Reynolds number for the Taylor instability. The two frequencies appear to be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears in the flow at R/Rc ≃ 12, where a weak broadband component is observed in addition to the sharp components; this flow can be described as weakly turbulent. As R is increased further, the component that appeared at R/Rc= 10·1 disappears at R/Rc = 19·3, and the remaining sharp component disappears at R/Rc = 21·9, leaving a spectrum with only the broad component and a background continuum. The observance of only two discrete frequencies and then chaotic flow is contrary to Landau's picture of an infinite sequence of instabilities, each adding a new frequency to the motion. However, recent studies of nonlinear models with a few degrees of freedom show a behaviour similar in most respects to that observed.