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Electronic fraud is highly lucrative, with estimates suggesting these crimes to be worth millions of dollars annually. Because of its complex nature, electronic fraud detection is typically impractical to solve without automation. However, the creation of automated systems to detect fraud is very difficult as adversaries readily adapt and change their fraudulent activities which are often lost in the magnitude of legitimate transactions. This study reviews the most popular types of electronic fraud and the existing nature-inspired detection methods that are used for them. The common characteristics of electronic fraud are examined in detail along with the difficulties and challenges that these present to computational intelligence systems. Finally, open questions and opportunities for further work, including a discussion of emerging types of electronic fraud, are presented to provide a context for ongoing research.