Insight processes that peak in “unpredictable moments of exceptional thinking” are often referred to as Aha! or Eureka moments. During insight, connections between previously unrelated concepts are made and new patterns arise at the perceptual level while new solutions to apparently insolvable problems suddenly emerge to consciousness. Given its unpredictable nature, the definition, and behavioral and neurophysiological measurement of insight problem solving represent a major challenge in contemporary cognitive neuroscience. Numerous attempts have been made, yet results show limited consistency across experimental approaches. Here we provide a comprehensive overview of available neuroscience of insight, including: i) a discussion about the theoretical definition of insight and an overview of the most widely accepted theoretical models, including those debating its relationship with creativity and intelligence; ii) an overview of available tasks used to investigate insight; iii) an ad-hoc quantitative meta-analysis of functional magnetic resonance imaging studies investigating the Eureka moment, using activation likelihood estimation maps; iv) a review of electroencephalographic evidence in the time and frequency domains, as well as v) an overview of the application of non-invasive brain stimulation techniques to causally assess the neurobiological basis of insight as well as enhance insight-related cognition. •EEG data suggest a primary role for alpha oscillations in prefrontal and parietal lobes.•fMRI data do not support overall right hemispheric dominance during Eureka moments.•Right temporal lobe shows concordant activations across fMRI and EEG studies.•Need for a consensus around validated insight tasks to be used for EEG/fMRI studies.•Future studies should test for causal role of brain areas by modulating brain oscillations.