Acura Integra 1993 (280,000 mls). I know it's normal for the fan to run for a few minutes after the engine is off, but mine lately has been running for up to 15 minutes or more. What are the most likely causes?

It may be time to replace the radiator, due to mineral buildup or corrosion.

If the radiator is in good condition, changing temperature sender, could help isolate.
Relatively cheap?

I will have the radiator checked. Others have suggested the coolant fan relay along with the temperature sensor. One thing I've noticed, during the fan's excessive run, if I start the car and turn it back off, sometimes the fan will then stop. Does that point to a more specific cause?

Sounds like the relay is "hanging" up.

I had it checked out and everything passed inspection. It seems it's doing what it's supposed to. The manual says the condenser fan can run for up to 1/2 hr and the radiator fan for 15 minutes after engine turn off. I live in the deep South. Still, I will continue to watch it.

The problem was the thermostat. Finally took it to the dealer as the indicator was climbing too close to red. It runs fine now.

If it happens again, and have to keep going, turn on the heater(it will usually be summer).
You can save your engine, just a little uncomfortable.