Creativity is defined quite simply as "the ability to create" in most lexicons, but, in reality, this is a complex and heterogeneous construct about which there is much to be discovered. The cognitive approach to investigating creativity recognizes and seeks to understand this complexity by investigating the component processes involved in creative thinking. The cognitive neuroscience approach, which has only limitedly been applied in the study of creativity, should ideally build on these ideas in uncovering the neural substrates of these processes. Following an introduction into the early experimental ideas and the cognitive approach to creativity, we discuss the theoretical background and behavioral methods for testing various processes of creative cognition, including conceptual expansion, the constraining influence of examples, creative imagery and insight. The complex relations between the underlying component processes of originality and relevance across these tasks are presented thereafter. We then outline how some of these conceptual distinctions can be evaluated by neuroscientific evidence and elaborate on the neuropsychological approach in the study of creativity. Given the current state of affairs, our recommendation is that despite methodological difficulties that are associated with investigating creativity, adopting the cognitive neuroscience perspective is a highly promising framework for validating and expanding on the critical issues that have been raised in this paper.