Fluctuating properties of the atmosphere, and in particular its water vapour content, give rise to phase fluctuations of astronomical signals which, if uncorrected, lead to rapid deterioration of performance of (sub)-mm interferometers on long baselines. The Atacama Large Millimetre/submillimeter Array (ALMA) uses a 183 GHz Water Vapour Radiometer (WVR) system to help correct these fluctuations and provide much improved performance on long baselines and at high frequencies. Here we describe the design of the overall ALMA WVR system, the choice of design parameters and the data processing strategy. We also present results of initial tests that demonstrate both the large improvement in phase stability that can be achieved and the very low contribution to phase noise from the WVRs. Finally, we describe briefly the main limiting factors to the accuracy of phase correction seen in these initial tests; namely, the degrading influence of cloud and the residual phase fluctuations that are most likely to be due to variations in the density of the dry component of the air