This subject comes up quite often here. Here's the thing; at least on models here in the U.S. (and I suspect certain design functions are the same world over) the cold air produced by the compressor unit only flows or is blown into the freezer compartment. For cold air to get into the fridge side it must pass through a vent system controlled by a thermistor in the fridge side.
Let me back up a minute... anytime the compressor is running (you can hear it) a device on most refrigerators is an evaporator fan in the freezer section which circulates the air throughout the whole appliance. That fan should be running whenever the compressor is running. When the compressor is running, open the freezer door, hold the interior light switch closed (should be on the door frame) and you should hear this fan running. If it's not running this can be the problem.
Next, if it is running,there's a damper door at the top of the wall dividing the two units through which cold airmoves from the freezer to the fridge that is usually controlled by a thermostatically controlled switch. This can fail and the door remain closed or nealry so causing the fridge side to be warm. On older units, this damper door is manually controlled when you set the thermostat on the fridge side... but even in that case the door may be hindered in opening.
Laslty make sure the freezer side isn't iced up, especially if it's a self defrosting side. This could mean the deice device(s) are inoperative. These can pretty easily be replaced by a reasonably astute DIYer...
Best of luck!