A dual-wheel steerable motorized caster has internal motors that are operated at different speeds to swivel a pair of wheels about a caster swivel axis. The motors are operated at the same speed to propel an apparatus to which the dual-wheel steerable motorized caster is coupled along an underlying surface such as a floor in a drive direction without swiveling the pair of wheels about the caster swivel axis. A patient support apparatus has one or more of such dual-wheel steerable motorized casters.