Ever since quantum mechanics was discovered nearly a century ago, famous scientists from Eddington to Wigner to Compton to Eccles to Penrose have speculated about possible connections to the brain -- a quest often parodied as "quantum mechanics is mysterious, the brain is mysterious, ergo they must be related somehow." In this talk, I'll offer a critical survey of these ideas from the modern standpoint of quantum information theory, pointing out the huge conceptual and experimental problems that have plagued most concrete proposals. However, I'll also explain why I think some role for quantum mechanics in cognition is not yet excluded, and discuss what sorts of advances in neuroscience and physics might help settle the question.