Psychologists studying problem solving have, for over 100 years, been interested in the question of whether there are two different modes of solving problems. One mode--problem solving based on analysis--depends on application of past experience to the problem at hand and proceeds incrementally toward solution. The second mode--problem solving based on insight--involves breaking away from what one knows, typically after one has reached an impasse, and entails solution occurring suddenly, in a leap beyond what one knows. A number of recent studies have used neuroscientific methods (e.g., fMRI; EEG) to demystify insight, i.e., to dispel what is seen as the mystery of insight in problem solving. This research has been summarized by Lehrer (2008), in the context of examination of a real-life example of insightful problem solving: R. Wagner Dodge's creation of an escape fire to avoid near-certain death from a forest fire. This article presents a critical review of neuroscientific studies of insight, which indicates that methodological flaws in those studies have precluded their furthering the understanding of insight. Tentative proposals are presented concerning the design of a study that would allow one to determine the brain states underlying insight. In addition, an analysis of Dodge's insight indicates that there may not be that much mystery standing in the way of our understanding what he did.