There is a company, Highlift Systems (liftport.com) whose mission is to build a space elevator. In June, 2002, they believed (or claimed to) they could accomplish the task in 15 years. In accordance with the writeups above, they primarily concentrated on length and quality in carbon nanotubes, and techniques of bonding them together.
For a base station, they picked a remote site in the middle of the Indian Ocean. Though initially ideas of a tall tower reaching up to meet a higher-starting cable were proposed, the plan did not involve one - it would be more expensive and dangerous than the cable itself.
The implementation under discussion would have made this a one-way elevator. That is, cargo would pull itself up, and then drop off the far end. Nothing would ever be let back down. This has the happy consequence that one can put multiple elevator cars on the same cable.
Also, no plans were in the works for a elevator car suitable for crew, let alone passengers.
Other companies are interested in the rotavator, which might be easier to build and would have some properties which might make it more useful.
Update, February 2009:
As one might have expected, snags have been hit and highlift systems is no more. It was succeeded by Liftport, which in 2007 shifted to focus on balloons.
Concrete efforts continue, though. In particular, there is a series of competitions sponsored by NASA, called "Elevator:2010" which is not promising an elevator by 2010, but to get an answer by then. The question is not really clear. One can follow all this at spaceward.org
Update, August 2012:
In late August, 2012, Liftport announced they had made a major breakthrough and could build a space elevator - for the moon - with existing technology, in 8 years. They have not been particularly clear on what the major breakthrough was. This is somewhat of a schedule slip from their claim in 2002, but surprisingly not too different. We shall see how that holds up at implementation time. Bizarrely, they launched a kickstarter for the paltry sum of $8k. This doesn't provoke much excess enthusiasm.
Meanwhile, spaceward.org lists dates within 2009 upcoming events. Suggesting going there for news wasn't so solid a notion, it seems.
Update, July 2015:
The site describes itself as a dangerous ideas factory. Hmm. Their strategic goals appear to be out of order, and appear to be most likely to be achieved by other people if they are. Several are dependent on advances in materials that are not very likely. I don't know who's going to build it first, but this makes me think it's not going to be liftport.