This is the first of two volumes of essays in commemoration of Alan Turing, whose pioneering work in the theory of artificial intelligence and computer science continues to be widely discussed today. A group of prominent academics from a wide range of disciplines focus on three questions famously raised by Turing: What, if any, are the limits on machine `thinking'? Could a machine be genuinely intelligent? Might we ourselves be biological machines, whose thought consists essentially in nothing more than the interaction of neurons according to strictly determined rules? The discussion of these fascinating issues is accessible to non-specialists and stimulating for all readers.