It is often impractical or dangerous to send people to explore underwater environments. In these situations, it is preferable to send robots such as autonomous underwater vehicles or remotely operated vehicles instead. Unfortunately, robots impart their own risks: they are typically made of rigid materials that can become lodged in confined spaces or harm underwater creatures and structures. Additionally, propellers or jet thrusters are typically used for propulsion, which are power intensive, have low efficiency, and impose additional concerns of entanglement and damage to their environment. Finally, they generate considerable noise and vibration, thus adding to the ambient noise pollution and disturbing sea life, preventing researchers from being able to study more timid animals. In this dissertation, I describe physical mechanisms to develop bioinspired, soft, swimming robots with an emphasis on actuation. First, I present an approach to use an arrangement of six artificial muscles based on dielectric elastomer actuators (DEAs) to actuate a tethered robot capable of anguilliform-inspired locomotion. Next, I demonstrate pulsatile, jellyfish-inspired locomotion using DEAs with a simpler actuation and control strategy, enabling an untethered, soft, swimming robot. Finally, I explore an alternative actuation approach to achieve more robust locomotion in a cephalopod-inspired robot based on slowly storing elastic energy and then quickly releasing it to eject a pulsed jet for propulsion. The first two robots are silent and use actuators that have a high energy density and efficiency, but provide low output power and swim at low speeds. In the cephalopod-inspired robot, we trade silence and efficiency for power and speed. These results demonstrate actuation strategies for realizing bioinspired locomotion in soft, swimming robots that could be useful for structural diagnostics, environmental monitoring, or search and rescue.