Motility of cells is at the root of many fundamental processes in biology: from sperm cells swimming to fertilize an egg cell, to metastatic tumor cells crawling to invade nearby tissues. 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. As a concrete example, we will report on reverse engineering of the euglenoid movement. The lessons learned in the context of swimming motility will be then applied to selected case studies of crawling motility.