I have a god-awful time with mountain bike tires. Luckily, I don't ride my mountain bike (a Trek 950, bought in '93) much any more.

I have found that this works best for me when dealing with road bike tires:

Put a little air in the tube, just enough to give it some shape, as others have already suggested.

work most of the tire back on, all but the last bit that won't go on. (Now the fun begins)

Hold the wheel against your stomach. Now, starting with your hands at 9 O'clock and 3 O'clock, work your hands right into the tire and rim, and work towards that troublesome last few inches of tire, while pressing the wheel against yourself. The objective is to build up as much "excess tire" as possible toward the end opposite your stomach. This can also be done in a sitting position, if you can find a nice comfortable bench.

Now, when you get to that list bit of tire, force it on with your thumbs. This may take repeated effort.
How easy, or difficult it is will vary with different tires. I'm running Conti Grand Prix's on my Gunnar. These don't flat often, but this technique has worked for me with these tires.

As already noted, you can use levers, if you work carefully. I'm always spooked of pinching the tire on the rim though. I'm told Pedro's levers work best for this. Anyone know anything about that?

The Kool Stop Beadjack looks like a great tool, if you can find them. There's also a product called a "Speed Lever" from Crank Brothers. Anyone have any experience with this tool? Park (of course,,) makes a professional tire mounting tool. I'll take a wild guess that shop mechanics, who are under time pressure, use them but won't admit to it.