The motivation for this book arose, in part, from a need for detailed replies to a number of queries and criticisms from readers of the author's previous book, The Emperor's New Mind , many of whom have gone to extraordinary lengths to avoid the conclusion that there must be something non-computational involved in thinking. Penrose searches for a means, within the constraints of the hard facts of science, whereby a scientifically describable brain might be able to perform the needed non-computational actions. He develops the argument of how quantum effects might have a fundamental relevance to consciousness and to non-computable brain action. Some of the more startling implications of this are based on new scientific information which is discussed in detail, leading to one of the most extraordinary and fundamental implications of quantum theory: quantum entanglement, the process by which the behaviour of `classically' described systems is explicable only in quantum mechanical terms. There is a thorough examination of the implications of G "odel's theorem maintaining that conscious thinking must indeed involve ingredients that cannot adequately be simulated by mere computation. No prior knowledge of quantum theory on the part of the reader is assumed. This book is intended for the general reader; physicists, computer scientists, philosophers, etc.