Many of our everyday activities, such as looking up information on the internet and journey planning, are supported by sophisticated algorithms. Some of our online activities are supported by the fact that we don’t have good algorithms for some problems: the encryption scheme that supports the privacy of credit cards in online transactions is believed to be secure precisely because there is no known fast algorithm for factoring large numbers. The Oxford Computer Science Professor explains a little of what we know about the limitations of algorithms, and also the famous P vs NP problem. This is the most important open problem in computer science and is one of the seven Millennium Problems of the Clay Mathematics Institute, which has offered a million-dollar prize for its solution.