A filament approximately 23,000 miles long would have a counterwight tethered in geosychronous orbit. This is the Clarke orbit proposed by Arthur C Clarke in 1945 and is where the majority of telecoms satellites are currently orbiting. This is a formidable challenge but steps are being made towards this goal. A prize has recently been won by a robotic climber rsing vertically to a height of 900 metres in under 2 minutes powered by a beam of light playing on solar cells. The climber does not need to reach escape volicity and can move up the filament at any speed. The material for the rope could be of carbon nanotubes or spider silk expressed in GM goats milk. Nothing has yet been made strong enough to support such extreme tension and weight but nothing should be ruled out. Since the shuttle has killed 14 astronauts  there is an ongoing need for a cheap way to get unmanned payloads into space.

The pros and cons of Space Elevators were thoroughly discussed in correspondence related to a New Scientist article on the subject in November 2008. Below are my responses to one of the postings.

Unfortunately, you'll have to seach the NS archive to find the questions to which the various enumerated points are my responses! However, the gist of it is that, in my opinion, the idea - although theoretically good - is impossible to implement, due to a considerable number of insuperable practical difficulties - including that mentioned in the last entry.

1. Coriolis forces are not illusory - they caused the Torrey Canyon oil tanker disaster.2. Nobody has said much about car propulsion, so here are a few thoughts. 2.1 If you use a laser beam to power the car, how are you going to stop the beam burning through the cable when it's blown by the wind or wobbles due to physical effects? 2.2 Apart from that, how will the power be used? If for an electric motor driving the car against the cable, the cable will be pulled down as much as the car goes up (thanks, Mr. Newton). 2.3 Granted, you could overcome this effect by thrusters on the topside station, but even with efficient ion drives, the thrusters still need "fuel" - mass for ejection - which would have to be obtained from somewhere. If it had to be brought up from Earth, this would severely reduce the efficiency of the system.2.4 The ion drives would have to be directed away from the cable (symmetrically opposed) to avoid damaging it, so that would also reduce the efficiency of the system.2.5 If you used ion thrusters on the car for lift to avoid cable vertical reaction effects, 2.4 would apply here too - and again, as in existing rockets, you would be lifting the "fuel".3. Yes, there is power to be had in both the upper atmosphere and Van Allen belts - but exactly how will you harvest it? By beefing up your conducting nanotube elevator cable? How much PD would you get across the height of an elevator car - enough to power it?4. My thanks to Sandy Henderson for setting out clearly how what I consider to be one of the more practical ways of getting material into space - an evacuated "Destination Moon" maglev rail gun - would be implemented. And, by the way, if you only used it for materiel, not people (leave them to rockets), you wouldn't be limited to 6G. Standard MIL Specs call for 50G, and if you stepped that up to 100G, you would probably only need a 50 mile track - quite practical. As far as I can see, the only problem would be the impact load of the hypersonic shockwave as the projectile leaving the vacuum tube hit the (even thin) atmosphere.5. The balloon idea looks interesting, but has a very limited load capability. On the other hand, I think a "grown-up" White Knight with an added SCRAMjets and Spaceship One-style pods could carry significant loads of both materiel and people into orbit much more economically than the present VTO rockets.6. Would anyone like to calculate the increase in global warming due to all the power burned in PCs, NASA's supercomputers and worldwide Internet servers from the theoretical work, discussion and other correspondence on this to my mind dead-end subject?

An orbital tower (which can have a "space elevator" running up and down it) is not a mass in geosynchronous orbit with a cable hanging off it down to earth.

It is a long thin structure in geosynchronous orbit and due to well understood orbital mechanics it's orientation is perpendicular to the earths surface.

If one starts building at 36,000km building in both directions one ends up with a tower 72,000km long with the other end travelling faster than orbital speed for that height.

A car trevelling towards 36,000km uses energy one travelling away generates energy. Having enough cars means means that the nett energy use is zero apart from efficiency losses.

Once built the costs of getting anything to orbit and beyond are reduced to negligable.

The great advantage to a space elevator is that the energy used to move it would be based on the ground (nuclear power facility) or in space (solar). This means that the shuttle would not need to carry its own fuel, thereby reducing weight by an incredible amount. This is important because the fuel and fuel canisters account for the vast majority of a modern shuttle's weight.

By reducing weight by such a large amount then less energy would be needed to lift the same weight into orbit as you would not have to use most of your fuel to lift the rest of your fuel. The great benefit to this is cost is reduced to such a large extent that it would be cheaper to use a space elevator than it would be to fly around the world.

As well as saving the need to carry a lot of fuel  as mentioned in other answers, the space elevator is far more efficient.  Rocket's waste most of their energy making hot blasts and noise which contribute nothing to the job in hand.  An elevator uses a cool and quiet electric motor, and that can be better that 90% efficient.