Space elevator is a proposed space transportation system, based on the idea of simply moving the freight between the surface of the Earth and Earth's orbit with the help of a long tether, connecting the Earth with a counterweighting mass, positioned above the geostationary orbit. A hoist (climber) would drive along the tether, prospectively powered by energy laser-beamed to it either from Earth or from the space. Such a setup is also applicable to other celestial bodies (Moon, Mars, ...). For a long time, no known material was strong and light enough for the construction of the tether (..of an Earth elevator). Recently, carbon nanotubes was found to have high enough tensile strength to weight ratio to be suitable for this purpose.
Space elevator reduces price of transporting each kg to space. Space elevator is potentionally much cheaper way of transportation to space than conventional rockets.
Space elevator can be scaled to increase access to space. Space elevator can attain much higher capacity than conventional rocket, allowing for access to space on a mass scale.
Space elevator is more environmentaly friendly than a classical rocket. Compared to conventional, chemical rocket engines, the climber could have zero environmental impact, when powered by solar energy from space.
Space elevator can be used more than once. Once built, space elevator could lift freight for many years, while classical rockets are one-use only. Even the Space Shuttle is only partially reusable (external tank is always lost during the launch) and requires major overhaul after each landing before it is ready for the next launch.
Space elevator is a more convenient way to get to space than a rocket. The acceleration (g-force) experienced in a space elevator can be adjusted to be near the natural value of 1g, as opposed to e.g. Space Shuttle, where it is up to 3g.
Oscillations of the tether could pose a fundamental problem for a space elevator. Apart from many other problems, severe lateral, longitudinal and torsional oscillations would probably develop on the tether (if only because of the lower 100km being slashed by winds), in a wide range of frequencies and such oscillations would eventually lead to its breakdown. The dampening of such oscillations could prove unsurmountable.