A vehicle speed control apparatus for allowing a host vehicle trailing a preceding vehicle to accelerate and decelerate without causing discomfort to the host vehicle driver. While the host vehicle is trailing the preceding vehicle, the apparatus on board the host vehicle stores the speed of the preceding vehicle entering a curve ahead as well as a headway distance of the host vehicle to the preceding vehicle. Upon reaching the curve, the host vehicle is controlled by the apparatus to enter the curve at a speed not in excess of the stored speed of the preceding vehicle.