Well that’s a hard question. There are alot of robots in the world (probably around 8 million), and although many of them are not very advanced it’s still hard to choose just one. Part of the problem is that you need to decide what you mean by advanced. Some robots are very mechanically advanced, others are mechanically simple but have more advanced artificial intelligence. Some are remotely operated by humans, and others are completely autonomous – in other words decide what to do entirely on their own.
I think I’m going to give you two answers. The first is a driverless car. You may not think it is a robot – but it is. This car – called Boss – won the 2007 DARPA urban grand challenge. It drove for 60 miles around a complex route – obeying all the traffic rules – while responding to other traffic and road users – completely autonomously. Really amazing. You can read about Boss – and the other grand challenge winners here: http://en.wikipedia.org/wiki/DARPA_Grand_Challenge
The other advanced robot is called Cronos and was designed and built by an old friend, Professor Owen Holland. Cronos looks scary because he’s humanoid, about 7 feet tall, and has one big eye in his head. Owen designed Cronos to investigate how robots with very complicated bodies can learn how to control themselves. Owen’s theory is that all complex bodies (robots and humans) need internal models of themselves and Cronos has a very sophisticated computer simulation of itself which acts as an internal model. This allows Cronos to learn how to control it’s incredibly complex – and floppy – body to do things like pick up a cup. (This may sound easy but really it’s not.) You can read about Cronos here: http://cswww.essex.ac.uk/staff/owen/machine/cronos.html
Could I build these robots? Maybe, but it would take alot of time, and money – and I would have to learn new advanced robotic techniques.