River bed scour is studied in the paper. Two types of scour are identified, namely local scour and fluvial bed scour. Fluvial bed scour, which occurs if the flow carries sediment load less than its capacity, can be further classified according to the causes as flood scour, downstream reservoir scour, hyperconcentrated flow scour and channelized river scour. The rate of scour is a key problem for understanding of scour process. More than 100 experiments with clear water flow and 33 experiments with sediment-laden flows were conducted to investigate the laws of scour rate. A scour rate formula, the first of its kind, is proposed in the paper, which is proved valid for unsteady flows. In sediment-laden flow the scour rate reduces in an exponential law following increase in the ratio of incoming rate of sediment load to the difference between the capacity and load of the flow. River bed inertia is a new concept of morphological dynamics which represents an important property of the river bed. The inertia of various bed compositions are calculated and the results illustrated that the inertia increases with sorting coefficient of bed materials. By employing the bed inertia and the Exner equation the sediment transport rate in the scour process is theoretically solved. The scour rate under various flow and incoming sediment load conditions can also be calculated from bed inertia. Calculated scour rates for natural rivers agree well with the measured data. A method for calculating maximum scoured depth of river bed during flood is suggested for hydraulic engineers to design safety depth of hydraulic works and pipelines across rivers.