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Biomimetic Flight
Biomimetics refers to the study of biological mechanisms in order to engineer machines that can mimic them. Here are some notable research projects that others have done in this field:

Flow Visualizations Around Real Insects
Insect Flight Group - flow visualization videos: dragonfly, Red Admiral butterfly, bumble bee, moth, locust
Dragonfly and Butterfly Flow Visualizations - by Robert Srygley

Artificial Insect Experiments
“Flapper” - Large-scale model of a moth by Charles Ellington at the Animal Flight Group. This was one of the first experiments to understand insect flight. Here is a explanation and a picture.
Robofly - Large-scale hoverfly wings flapping in oil. See this article and video presentation by Michael Dickinson.
Insect-Based Flapping Wing MAV - by Gordon Leishman

Computational Simulations (CFD)
Insect Flight - Jane Wang made one of the first simulations to predict that an insect can produce sufficient lift to remain aloft. Here is another article about it.
Vortex Method Simulations - by Jeff Eldredge
Simulation - by Rajat Mittal, Here is another animation.
Bat Flight Simulation - Nice visuals but low Reynolds number.

Other Biomimetic Robots
Snake Robot - By Gavin Miller. This is one of my favorites. The movement appears very smooth on video. See also the sidewinding snake. Here is another robotic serpent by NASA.
Spinybot II - robot that can climb walls
RoboPike - a robotic fish
robotic caterpillar
RoboSnail - Here’s an unusual idea.
Robotic Mule
RoboStrider - modelled after a water-strider
RoboRoach - By Roger Quinn. This hexapod is modelled after a cockroach.
robotic dolphin
RoboLobster - by Joe Ayers