The idea of robots taking over the world seems a bit absurd if you look at technology in the sense of the last century. The idea of losing control of our technology seems like something from Frankenstein, more more modern versions such as the matrix and Terminator but this possibility becomes a bit more real feeling as you begin to understand the exponential growth of technology in this century, the new ways that we are using technology and the nature of humans.
The first step in understanding this as a possibility is understanding Moore's law. This law states at its most basic that computer hardware will double in power every two years and it has been remarkably accurate except that so far the doubling has been every 18 month not two years. In the early ages of computers this allowed computers to move up from text to visual input, and allowed programming to become bloated. The assumption among industry experts is generally that this expansion can't continue forever. The speed of light if nothing else will halt progress on computer speed, but if you take technology in general you can see that this trend is not simply one of computers. The industrial age was shorter than the classical age, the information age shorter than the industrial and the digital age is likely to be even shorter than that.
Most pertinent to this conversation though is that this doubling of computer power will mean that by 2020 computers able to process as much information as the human brain will be not only available but inexpensive and will likely continue to double every 18 months for some time after that.
This though does not mean that computers will rule us. There is more to human brains than simple computational power. Instead it is likely that the trends of how we use computers will simply continue as they are now, so how do we use computers now? Most of us find information, watch entertainment and play games, but these are not the only uses for computers and the others are becoming more common. Including, monitoring people, work in factories, science and combat. With these trends continuing we could find that humans no longer need to do many of the things that we have in the past, but that alone will not mean we are being controlled by robots, only that they will fight our wars, discover new science and build things for us. One can hardly be thought to be ruled by his robot butler.
Finally we come to the most disturbing part of this idea, the nature of humans. Humans have since the beginning of time broken into the idea of followers and leaders. This trend is unlikely to stop soon, but the ability for men to hold power has in the past always depended on the agreement of other men, either those in the citizenry or at least those in the military. With the growth of the power of the robot this need for others is diminishing. Each generation of technology gives more power to all humans and that power if not watched will be abused, so while it is unlikely that computers alone will rise up and take over the world it is possible that at some point men will use robots to attempt to control other men. This may not truly be robot rule as a handful of humans may control those robots, but it is at least as dangerous.