An electric power assisted steering apparatus for a vehicle, comprises a steering mechanism, which operatively connects a steering wheel to the road wheels of the vehicle, an electric motor operatively connected to the steering mechanism, a torque sensing means adapted to produce a first output signal indicative of the torque carried by a portion of the steering mechanism, a vehicle speed sensing means for producing a second output signal indicative of the speed of the vehicle, a signal processing unit adapted to receive the first and second signals and to produce a torque demand signal representative of a torque to be applied to the steering mechanism by the motor; and a motor drive stage adapted to provide a drive current to the motor responsive to the torque demand signal. The apparatus includes torque limiting means, arranged to limit the magnitude of the torque to be applied to the steering mechanism to a maximum of a value that increases in time from a first value to a second value at a rate that is dependent on the second signal. Additionally or alternatively, it may be arranged to limit the maximum torque to a value that increases from a first to a second value at a rate that varies with time.