Just picked up a 1978 vette. It's my first classic car, so perhaps this kind of thing is normal.

After driving for about 20 minutes, I notice it starts to get warm near my feet. When I touch the hood release shaft it is very warm. The temperature gauge is broken, but is being fixed by the dealer I bought it from next week, so I can't tell if the engine is running hot or overheating.

I have stopped and checked the engine when I noticed it, no funky smells or anything, though, the engine is obviously warm.

Am I in danger of it overheating?

Is this type of thing normal in cars of that era?

This is normal. I have an 05 and does the same thing. Just a Corvette thing I guess. It nice in the winter time though:grinno:

Yes this is pretty much normal with these cars. I own a 1972 Corvette and i bypassed the heater hose and installed a thermal padding between the floorboard and the carpeting. The fiberglass body absorbs the engine heat and it passes it on to you and your passenger. What i done to my vette reduced the heat drastically, but it still gets pretty warm inside too. Do you have A/C?