We will discuss the mechanical bases of cellular motility by swimming and crawling. Special emphasis will be placed on the connections between low Reynolds number swimming and Geometric Control Theory, and on the geometric structure of the underlying equations of motion.

We will then examine some concrete example, taken from the case studies that have been recently considered by our group and including: reverse engineering of the euglenoid movement, undulatory locomotion of snake-like robots, and one-dimensional models of slender crawlers.

Finally, we will re-examine the lessons learned in the context of biological cell motility with the aim of building a dictionary of elementary motility mechanism to be used in prototypes of bio-inspired motile micro-robots.