Current Micro Aerial Vehicles (MAVs) are greatly limited by being able to operate in air only. Designing multimodal MAVs that can fly effectively, dive into the water and retake flight would enable applications of distributed water quality monitoring, search and rescue operations and underwater exploration. While some can land on water, no technologies are available that allow them to both dive and fly, due to dramatic design trade-offs that have to be solved for movement in both air and water and due to the absence of high-power propulsion systems that would allow a transition from underwater to air. In nature, several animals have evolved design solutions that enable them to successfully transition between water and air, and move in both media. Examples include flying fish, flying squid, diving birds and diving insects. In this paper, we review the biological literature on these multimodal animals and abstract their underlying design principles in the perspective of building a robotic equivalent, the Aquatic Micro Air Vehicle (AquaMAV). Building on the inspire-abstract-implement bioinspired design paradigm, we identify key adaptations from nature and designs from robotics. Based on this evaluation we propose key design principles for the design of successful aerial-aquatic robots, i.e. using a plunge diving strategy for water entry, folding wings for diving efficiency, water jet propulsion for water takeoff and hydrophobic surfaces for water shedding and dry flight. Further, we demonstrate the feasibility of the water jet propulsion by building a proof-of-concept water jet propulsion mechanism with a mass of 2.6 g that can propel itself up to 4.8 m high, corresponding to 72 times its size. This propulsion mechanism can be used for AquaMAV but also for other robotic applications where high-power density is of use, such as for jumping and swimming robots.