A milking system includes one or more milking devices and a control unit. The system is provided with a plurality of milking cups, a first milk jar for receiving the milking, a first pumping device for pumping the milking, a second milk jar for receiving the milking from the first milk jar, a second pumping device for pumping at least a part of the milking, and at least one storage tank for storing at least a part of the milking. The second pumping device pumps the milking at a lower flow rate than that at which the first pumping device pumps the milking. As a result, the milk is for the most part moved with low mechanical load and little admixture with air which improves milk quality. In the case of a plurality of milking devices in the milking system, various milkings can be moved and treated separately.