Jellyfish uses jet propulsion to achieve a diversity of propulsion modes in the water. In this article, a miniature jellyfish-inspired swimming robot is designed and built, which is capable of executing horizontal and vertical propulsion and maneuvers. In order to imitate the jellyfish in terms of morphology and kinematics, the robotic jellyfish is designed to be comprised of a streamlined head, a cavity shell, four separate drive units with bevel gears, and a soft outer skin encasing the drive units. A combination of four six-bar linkage mechanisms that are centrally symmetric is adopted as the driver to regulate the phases of contraction and relaxation of the bell-shaped body. Furthermore, a triangle wave generator is incorporated to generate rhythmic drive signals, which is implemented on the microcontroller. Through independent and coordinated control of the four drive units, the robotic jellyfish is able to replicate various propulsion modes similar to real jellyfish. Aquatic tests on the actual robot verify the effectiveness of the formed design scheme along with the proposed control methods.