I've answered this question a number of times,which I'm glad to do, but I don't wish to be uneccessarily redundant.
Since you say the freezer side is working fine,I'd have to respectfully disagree with Biggles.. I'm relying on my knowledge of fridge/freezers here in the U.S., but have found that most share similar construction.
Most peope aren't aware that the compressor output affects only the freezer side. It cools the air to a thermostatically determined temperature. Then an air flow device allows cold air to move to the fridge side. It's associated with a blower and the door on the devicee is usually controlled by a thermostat in the fridge side. Some less expensive models actually control the opening and closing of this door like device manually. But, let's assume yours is thermostatically controlled. Either the thermostat has failed or the door device, usually operated by a small electric motor or solenoid has failed.
Try this; when the compressor is running (you should be able to hear it) open the fridge door (I'm assuming it's a side-by-side) and find the switch in the door frame that shuts of the fridge light. Press it in and see if anything moves on the cold air flow door or if you can hear the fan, which is often also controlled by the light-off switch. My guess is the device itself has failed. It can be replaced fairly easily once you've secured the correct part number.
Often, you can go online and Google-search the manufacturer and find a troubleshooting guide as well as a parts list.
As an aside; if yours happens to be a freezer on top or bottom instead of side-by-side, there should be shuttle door for the cold air circulation...
Best of luck!