Railway rail fastening apparatus, for suspending a railway rail at discrete locations along its length above a rail foundation, comprises two rail suspending assemblies located when in use one on each side of the rail. Each assembly comprises an elastic member for supporting the head of the rail on one side thereof and a bracket for applying a lateral clamping force to the elastic member. The rail suspending assembly on one side of the rail is located such that the centre line of the assembly is offset with respect to that of the rail suspending assembly on the other side of the rail by x mm along the longitudinal axis of the rail, where x is 50≧x>0 mm.