Prof. Margaret Boden (Philosophy, Sussex) delivers a lecture as part of the Keble College Creativity series. Creativity is likely to remain a neuroscientific mystery for many years. Of the three types of creativity (combinational, exploratory, and transformational), only the first has been significantly illuminated by neuroscience. And even that is not fully understood in neural terms. The other two are even more recalcitrant. This is due to difficulty in defining thinking styles in art or science, and in identifying the various computational processes that are involved in using them. Without doing that, helpful neuroscientific questions simply cannot arise. One key problem is that hierarchical systems -- including many creative "styles"-- cannot yet be effectively represented by (connectionist) computer models inspired by the neural networks in the brain. Another is the difficulty of explaining the recognition of relevance in computational/theoretical terms. Creative Commons Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/