Recent computers are immensely more powerful than the older ones, but this power comes at a price. See, the faster it works, the more power it uses, and the more it makes heat. Heat is bad for electronics. They don't like it.
So, in the past, first the heat sink was invented, then the computer fan. The heat sink is a piece of irregular metal that dissipates the heat into the air by virtue of its spiky, semi-irregular face. Metal conducts heat really well. Air, not so much. The fan blows air on the heat sink, taking some of the heat with it. At the cost of noise and a little power. Your computer is noisy to the degree that it is because of the fans. The fans also tend to get louder with age, and my machine is starting to sound like a jet taking off. A jet with what sounds like a desperate need to use the little jet's room.
Some people would like their computers to go even faster, so fast that fans just aren't going to work anymore. Now what? They use water cooling. Water cooling works like your car's radiator. Water goes through pipes that touch the heat sink, taking heat with them. Water isn't very good at collecting heat, but it absorbs a lot in aggregate. Then, you run the hot water through a radiator, which uses the heat-sink and the fan technologies to dissipate the heat. This way, you can use just one big, quiet, reasonably efficient fan instead of the four or five normally used, and it's quiet. Although if anything leaks, you will experience the mother of all short circuits.
But let's say you don't like that, and you want to do it the weird way? Fine, Peltier cooling. You attach a big chunk of metal to the hot parts, and plug it in. The metal becomes freakishly cold on one end, and mind-bogglingly hot on the other. The hot end is even hotter than what the cold end was before cooling. You then cool off the hot end however. Yeah, a fan if it's inside the case, but if it's on the outside, you could drip water on it. (Evaporative cooling? Very effective indeed.) Peltier cooling is not very efficient, but it is perfectly silent, which is why it's in digital cameras and whatnot.
Speaking of evaporative cooling, you can already buy devices that are a long metal tube the bends upwards. There is liquid sealed into the tube. When exposed to heat, the liquid evaporates, absorbing the heat into itself, and rising up the tube as a vapor. At the end of the tube is a radiator, which cools the vapor until it liquefies again. The now liquid flows back to the bottom, ready to absorb more heat. This only works if the radiator is above the heat source, and useless in zero-g. (But when are we ever going to need to run electronics there?)
Okay, you can buy all of this at the store. Now how about crazier techs?
My first idea is central-vacuum cooling. Many homes and businesses are built with a pneumatic tube system that connects to an air-pump that sucks the air from the tubes. This is done so that any hose connected is now a vacuum cleaner. The system runs quietly all the time, and uses the same amount of power despite being much more powerful. Given such a system, connect the computer case to it. Air will now be drawn in through any holes in the case by the low pressure, dropping temperatures inside. As old, heat-bearing air is sucked away, fresh air from the room replaces it.
Speaking of gases, how about a compression cycle? We run freon tubes in a maze inside and outside the case. Before it goes outside, we compress it with a compressor. It radiates its heat to the outside world. Now as it returns to the inside, we expand it again. The computer case is now a ridiculously cold freezer. (This is how your refrigerator works.)
Or, if you're totally insane, how about a cooling tower? A small amount of water is drawn off from a tap, run through a water-cooling system, but instead of a radiator, it is put in a metal tower at the top, where it evaporates into steam. As an added bonus, such a computer doubles as a humidifier for those awful, dry climates.