This paper reviews work on the stability of a particulate stream bed to the erosive attack of a flow with a free surface. Attention is given to the development of the hydraulic, the potential-flow and the rotational-flow models, and to the roles of phase lags and transport laws. The relationship of the stability theory to the ultimate form of the stream bed is discussed, and the current level of understanding of bed features is examined. Some investigations relating to a wider range of erosive processes are noted, and an extensive bibliography is provided.