Geostationary orbital tethers
This concept, also called an orbital space elevator, geostationary orbital tether, or a beanstalk, is a subset of the skyhook concept, and is what people normally think of when the phrase 'Space elevator' is used (although there are variants).
Construction would be a vast project: a tether would have to be built of a material that could endure tremendous stress while also being light-weight, cost-effective, and manufacturable in great quantities. Today's materials technology does not meet these requirements, although carbon nanotube technology shows great promise. A considerable number of other novel engineering problems would also have to be solved to make a space elevator practical. Not all problems regarding feasibility have yet been addressed. Nevertheless, the LiftPort Group believes that the necessary technology might be developed as early as 2008 and that by developing the technology, the first space elevator could be operational by 2014.