We present several biologically inspired concept designs and feasibility analyses for underwater swimming exoskeletons. These designs are biologically inspired, based on observations of dolphins, sea turtles, and penguins. Biologically inspired designs have the advantages of stealth, maneuverability, and a natural interface when compared to propeller driven underwater propulsion devices. We present a lower body concept, based on dolphin locomotion and an upper body concept based on sea turtle and penguin locomotion. The dolphin based concept has the advantages of using the wearer's most powerful muscle groups, of being the most natural to swim with, and of leaving the user's hands free for other tasks. The sea-turtle based concept has the advantage of novelty and may have appeal as a recreational device. We predict that with actuation and energy storage components available today, an exoskeleton that produces a cruising speed of over 1 m/s and a top speed of over 1.5 m/s is feasible. We estimate that cruising at 1 m/s can be achieved with less than 504 Watts power consumption, which translates into 2.4 kg of off-the-shelf silver-zinc batteries per hour of operation.