This article has two goals. First, the ideas outlined here can be seen as a sustained and disciplined demolition project aimed at sanitizing our bad habits of thinking about creativity. Apart from the enormous amount of fluff out there, the study of creativity is, quite unfortunately, still dominated by a number of rather dated ideas that are either so simplistic that nothing good can possibly come out of them or, given what we know about the brain, factually mistaken. As cognitive neuroscience is making more serious contact with the knowledge base of creativity, we must, from the outset, clear the ground of these pernicious fossil traces from a bygone era. The best neuroimaging techniques help little if we don't know what to look for. Second, as an antidote to these theoretical duds, the article offers fresh ideas on possible mechanisms of creativity. Given that they are grounded in current understanding of cognitive and neural processes, it is hoped that these ideas represent steps broadly pointing in the right direction. In the end, the fundamental question we must ask ourselves is what, exactly, are the mental processes--or their critical elements--that yield creative thoughts.