The greatest challenge, by far, is the cable itself. It must be made of such a material, which must not only be able to hold the weight of the elevator (which is a trivial issue), but also be able to sustain its own weight over such a great length (which is extremely difficult).
For this purpose the cable must be properly tapered, i.e. widest at the point of maximum stress, the geostationary orbit and narrowest at the base. But practically this is not possible. Say we use steel. If the cable is 1 millimeter across at the base then it must be 40 billion km in diameter at the GEO in order to sustain its own weight.
That is because a tensile strength of 62.5 gigapascals is required.
The only answer is carbon, which in the form of diamond shows unbelievable mechanical properties. It is true that diamond can't be spun into filaments, but there is a form of carbon that combines strength with length: nanotubes. These tiny, hollow cylinders made from sheets of hexagonally arranged carbon atoms exceed the tensile strength of steel by at least a factor of 100. The tensile strength of nanotubes is 130 gigapascals, which surpasses the required strength by a safe margin.
But there are some other problems too. Diamond, an extremely costly substance comes at an astounding price of $500 per gram and is also scarcely available. This cost of making nanotubes can be reduced greatly by using newer techniques like making them using laser vaporization of graphite.
Next option could be of raising the base of the cable by a few tens of kilometers. The girth of the cable at the GEO would reduce, saving us tons of the material.
For this we could locate the base at the top of a mountain in the equatorial region, where torpedoes, storms and whirlwinds are scarce. But this clearly spells doom in the incidence of the cable snapping. So the idea of erecting a tower a few ten kilometers tall is being considered which is also not feasible, as the means of getting people to the top of the base will steeply increase the cost of the trip to space.
Along with nanotechnology, tether technology (see propulsion) and electro-magnetic propulsion (see electro-magnetic propulsion) are also being worked upon for this project.
There are still many unanswered questions about this technology as of today. The project of space elevators has surely grown out of infancy a long time ago, but it still may be half a century or more before it becomes a reality.
For more information on Nanotubes please click here