Keywords: Low Reynolds number; locomotion; symmetry-breaking; synchronization; cilia; optimization Abstract: Fluid mechanics plays a crucial role in many cellular processes. One example is the external fluid mechanics of motile cells such as bacteria, spermatozoa, algae, and essentially half of the microorganisms on earth. The most commonly-studied organisms exploit the bending or rotation of a small number of flagella (short whip-like organelles, length scale from a few to tens of microns) to create fluid-based propulsion. As a difference, Ciliated microorganisms swim by exploiting the coordinated surface beating of many cilia (which are short flagella) distributed along their surface. In this talk, we consider two instances of symmetry-breaking arising in small-scale locomotion. First, we address the observed flagellar synchronization between eukaryotic cells swimming in close proximity. By using a two-dimensional model, we show analytically and computationally that synchronization between co-swimming cells can be driven by hydrodynamic interactions alone if there is a geometrical symmetry-breaking displayed by the their flagellar waveforms. In a second part, we pose the problem of ciliary propulsion as an optimization problem. Specifically, for a spherical body, we compute numerically and theoretically the time-periodic tangential deformations of the body surface which leads to swimming of the body with optimal hydrodynamic efficiency. We show that this calculation leads to symmetry-breaking in the surface actuation, and the emergence of waves, reminiscent of the metachronal waves displayed by real biological cilia.