I've been meaning to put something up here about the Space Elevator for years. During that time most people seem to have somehow heard about this rather unusual but brilliant idea. At least most of the ones I know, which is, of course, a fairly unusual crowd in its own right.
The Problem: getting to space. The Solution: climbing a rope. All you have to do is tie a counterweight to the end, and the centripetal force keeps it in geostationary orbit. Then, instead of worrying about the inefficiency and danger of getting into space with rockets, you just have to worry about the inefficiency and danger of getting into space by going up a ~22,000 mile tether.
You would locate the base at the equator and devise some sort of contraption capable of pulling itself up. Although this is certainly harder than launching a rocket, once you figured it out, this method would be a far superior way of getting heavy payloads into space. And not only low-Earth orbit; if desired you could use the rotational velocity to shoot off into space like a slingshot (a rare occasions when the word "slingshot" could be used in a non-idiotic sense in reference to space travel). Only twice a day, your tangential velocity would be in the plane of the solar system, (because of the Earth's tilt) and the directions within the plane would, of course, change over the course of the year. Apparently, one of the main ideas for powering the climber involves transferring energy to it from a massive laser beam at the base. I'm not sure how they would manage to avoid dispersion in the atmosphere, but maybe I've just been looking at stuff written by the wrong people. The other undeveloped technology required is the material for the tether, which has to be very strong and very light, and capable of being made continuous for hundreds of miles. Carbon nanotubes seem to be the popular choice here, since anytime you don't know the answer to something, ponderously replying with the phrase "carbon nanotubes" tends to shut down all further discussion.
Nevertheless, solving the many issues involved in creating a successful space elevator is far less interesting than considering all the ways the project could go wrong. The cable could snap, flinging the counter-weight (probably some kind of space station) off into the solar system at a high rate of speed -- possibly with the payload as well, the falling cable creating an equatorial swath of destruction. Probably, if light and made out of carbon, a lot of it would burn up in the atmosphere, but I can imagine a scenario where it snaps and is then dragged along the ground at an extremely high rate of speed (since the counter-weight of a taut cable would lag behind the ground base). That would certainly be fun. It is also hard to see how you could protect a cable that leads into space from sabotage, considering that it rises vertically for the entire atmosphere. Best of all, collisions with satellites are not only possible but inevitable given enough time. A line cutting upwards from the equator will be intersected by any normal, unaltered orbit eventually.
NOVA has a nice segment going into the basics and showing a rather unusual contest that took place recently to make a solar powered climber. You can watch it here for free.
[And if that isn't enough, there are a disturbing number of CGIed clips demonstrating the wonders of the Space Elevator over techno music or narration by true believers on YouTube. Though I wouldn't recommend any of them.*]
*Update: except for any produced by our commenter from LiftPort, which are excellent.