Science in the twenty-first century inherits a fantastic record of accomplishment. Yet there remain enormous questions to answer - some of them revealed by those very past successes. Why is 95 per cent of the universe undetectable 'dark matter', and does it control the cosmos - how it began, its eventual fate and whether it is finite or infinite? Is our universe the only one, or could there be an infinity of others? Do other universes operate on different physical laws. Is there complex life in other star systems. If so, how far could it outstrip our intelligence? Could a sufficiently advanced life form even create new universes to its own design? In this playful and accessible book, Britain's best-known commentator on the most profound questions in science looks at whether they could be answered and asks, if they are, will the human brain be capable of comprehending the results? What we Still Don't Know looks beyond the edge of current scientific understanding, and poses problems for a new generation of inquiry.