Could be a blockage in the fridge drainage hole at the base of the cooling panel at the back of the fridge. To check this, switch off the fridge and open door for half an hour. Remove the plug of ice from the V-shaped gulley at the back and see if water drains through properly to the outlet at the back of the fridge, at the bottom. That pipe will probably drain into a plastic open topped tank that sits on top of the compressor (the thing that throbs when it's switched on. The tank is put there because the heat from the compessor evaporates the overflow water) If the tank is bone dry, the chances are the pipe above it is blocked. Get a packet of old fashioned pipe cleaners, make a chain of them and rod down through the outlet pipe inside the fridge until it emerges out of the bottom.
Another explanation - on auto-defrost machines- may be that the thermostatic relay, that switches the cooling panel on and off to allow the fridge freezer to drain, is broken. It depends whether you have a fridge freezer where both parts have independent motors (the more expensive kind). If they're both linked to the same motor, that would explain why your freezer is icing up too. For that you'll have to get an engineer in. But if you tell them what's needed and ask for a quote on the phone, the chances are you won't be ripped off.
Good luck
Joe, October 2007