Several physico-mechanical designs evolved in fish are currently inspiring robotic devices for propulsion and maneuvering purposes in underwater vehicles. Considering the potential benefits involved, this paper presents an overview of the swimming mechanisms employed by fish. The motivation is to provide a relevant and useful introduction to the existing literature for engineers with an interest in the emerging area of aquatic biomechanisms. The fish swimming types are presented, following the well-established classification scheme and nomenclature originally proposed by Breder. Fish swim either by body and/or caudal fin (BCF) movements or using median and/or paired fin (MPF) propulsion. The latter is generally employed at slow speeds, offering greater maneuverability and better propulsive efficiency, while BCF movements can achieve greater thrust and accelerations. For both BCF and MPF locomotion, specific swimming modes are identified, based on the propulsor and the type of movements (oscillatory or undulatory) employed for thrust generation. Along with general descriptions and kinematic data, the analytical approaches developed to study each swimming mode are also introduced. Particular reference is made to lunate tail propulsion, undulating fins, and labriform (oscillatory pectoral fin) swimming mechanisms, identified as having the greatest potential for exploitation in artificial systems.