This paper investigates the three-dimensional (3-D) maneuverability of a gliding robotic dolphin through analyzing the horizontal and vertical motions. Concretely, in order to improve the performance in the horizontal plane, the gliding robotic dolphin is developed with a rudder installed on its belly, and the 3-D full-state dynamic model is derived briefly. By setting two typical body and/or caudal fin (BCF)-based and three typical median and/or paired fin (MPF)-based turning patterns, the turning performances are revealed and analyzed. With regard to the vertical plane, a novel finite state machine framework is proposed to realize the vertical downward action. Finally, extensive experiments involving the turning and vertical motions are conducted to illustrate and analyze the 3-D maneuverability, which provides valuable guidance for the real-world applications of the gliding robotic dolphin.