  Cilia and flagella are actively bending slender organelles, performing
functions such as motility, feeding and embryonic symmetry breaking. We review
the mechanics of viscous-dominated microscale flow, including time-reversal
symmetry, drag anisotropy of slender bodies, and wall effects. We focus on the
fundamental force singularity, higher order multipoles, and the method of
images, providing physical insight and forming a basis for computational
approaches. Two biological problems are then considered in more detail: (1)
left-right symmetry breaking flow in the node, a microscopic structure in
developing vertebrate embryos, and (2) motility of microswimmers through
non-Newtonian fluids. Our model of the embryonic node reveals how particle
transport associated with morphogenesis is modulated by the gradual emergence
of cilium posterior tilt. Our model of swimming makes use of force
distributions within a body-conforming finite element framework, allowing the
solution of nonlinear inertialess Carreau flow. We find that a three-sphere
model swimmer and a model sperm are similarly affected by shear-thinning; in
both cases swimming due to a prescribed beat is enhanced by shear-thinning,
with optimal Deborah number around 0.8. The sperm exhibits an almost perfect
linear relationship between velocity and the logarithm of the ratio of zero to
infinite shear viscosity, with shear-thickening hindering cell progress.
