A supercharging arrangement for an internal combustion engine comprising: a supercharger having a first rotational drive input shaft; a transmission having a second rotational drive input shaft to receive drive from the internal combustion engine via a pulley at a speed ratio, and a rotational drive output shaft connected to the first rotational drive input shaft of the supercharger, wherein the transmission includes a variator operatively connected between the second rotational drive input shaft and the rotational drive output shaft of the transmission, which variator has an output shaft that is driven at an operating ratio (Rv) from a third input shaft coming from a fixed ratio stage of the transmission; and a control system that operates to cause the engine to deliver an amount of torque in response to a state of an operating control, wherein the control system is operative to set the operating ratio (Rv) of the variator to a variator ratio request and to adjust the operating ratio (Rv) of the variator to ensure that a rate at which the supercharger speed changes does not exceed a predetermined limit, and to ensure that the torque that is applied to the output of the engine or to the variator does not exceed a threshold, and wherein the control system is operative to calculate an air pressure at an outlet of the supercharger to generate a supercharger air pressure request based on the state of the operating control, calculate a mass air flow required to achieve the supercharger air pressure request at a current speed of the internal combustion engine, determine a speed of the supercharger to achieve the calculated mass air flow, generate a supercharger speed request based on the determined speed of the supercharger, and calculate the variator ratio request based on the supercharger speed request. a supercharger having a first rotational drive input shaft; a transmission having a second rotational drive input shaft to receive drive from the internal combustion engine via a pulley at a speed ratio, and a rotational drive output shaft connected to the first rotational drive input shaft of the supercharger, wherein the transmission includes a variator operatively connected between the second rotational drive input shaft and the rotational drive output shaft of the transmission, which variator has an output shaft that is driven at an operating ratio (Rv) from a third input shaft coming from a fixed ratio stage of the transmission; and a control system that operates to cause the engine to deliver an amount of torque in response to a state of an operating control, wherein the control system is operative to set the operating ratio (Rv) of the variator to a variator ratio request and to adjust the operating ratio (Rv) of the variator to ensure that a rate at which the supercharger speed changes does not exceed a predetermined limit, and to ensure that the torque that is applied to the output of the engine or to the variator does not exceed a threshold, and wherein the control system is operative to calculate an air pressure at an outlet of the supercharger to generate a supercharger air pressure request based on the state of the operating control, calculate a mass air flow required to achieve the supercharger air pressure request at a current speed of the internal combustion engine, determine a speed of the supercharger to achieve the calculated mass air flow, generate a supercharger speed request based on the determined speed of the supercharger, and calculate the variator ratio request based on the supercharger speed request.