From the largest animals to the smallest single-cell organisms, motility is a fundamental characteristic of life. Cell swimming, in particular, underpins a wide range of fundamental biological phenomena, including microbial grazing that fuels the base of the food webs, inter-microbial communication, animal reproduction, and parasitic infection—including severe diseases like malaria trypanosomiasis. Motile microorganisms are among the most important life-forms on earth, not only because of their abundance, but also because of their vital functions, e.g., in symbiosis with mammals or in ecosystems. Unravelling the basic principles of their propulsion mechanisms is essential for the development of novel strategies in the treatment of diseases, to understand microbial transport like the migration of marine phytoplankton in aquatic environments, and ultimately to open avenues for control of biological systems and the design of artificial nanomachines.