Powerful arguments can be given, to support the case that the quality of human understanding is not something that can be simulated in a trustworthy way, by any entirely computational system. If this case is accepted, it raises the question of what deep physical processes and what subtle brain structures might be involved in order that consciousness can come about. Some remarkable new observations concerning A-lattice microtubules will be briefly described, these having considerable relevance to this issue.Sir Roger Penrose is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. He has received a number of prizes and awards, including the 1988 Wolf Prize for physics which he shared with Stephen Hawking for their contribution to our understanding of the universe. He is renowned for his work in mathematical physics, in particular his contributions to general relativity and cosmology. He is also a recreational mathematician and philosopher.