Comprehension of natural language--stories, conversations, text--is very simple for those doing the comprehending and very complex for cognitive neuroscientists. It also presents a paradox: the advantage of the left hemisphere (LH) for most language tasks is one of the best-established facts about the brain; yet, when it comes to comprehending complex, natural language, the right hemisphere (RH) might play an important role. Accumulated evidence from neuropsychology, neuroimaging, and neuroanatomy suggests at least three roughly separable (but highly interactive) components of semantic processing. Each process in turn has bilateral components, with the RH component performing coarser computations for the same general process. Examining asymmetrical brain and cognitive functions provides a unique opportunity for understanding the neural basis of complex cognition.