There is a slow, steady drip coming from the top of the inside of the refrigerator. It is enough to put water on all the shelves. Right now I have bowls inside to catch the water.

Is there anything I can do or should I all an applaiance reprair company?

The refrigerator is about ten years old and never had problems.

As a lay person in appliance repair who also likes to figure out why something goes wrong with an appliance and then, if I can, fix it myself, the first thing that I would check would be to see if all the seals/rubber around the doors were tight against the body of the refrigerator. It could be defrosting itself because warm air is getting inside?

That would be how I would start the investigation into the problem.

We do have some people on this site who seem to be experts in appliance repair. And, I'm hoping that one of them will notice your post.

You probably have a plugged up drain line. If your freezer is above your refrigerator there will be a drain that runs down to the condenser where the water is evaporated. If it's a frost free, there will be about two or three defrost cycles each day when the compressor shuts of and a defrost heater will come on and melt the ice off you freezer evaporator coil. If the drain is plugged it has no place to go except down into your refrig.