  The locomotion of microorganisms and spermatozoa in complex viscoelastic
fluids is of critical importance in many biological processes such as
fertilization, infection, and biofilm formation. Depending on their propulsion
mechanisms, microswimmers display various responses to a complex fluid
environment: increasing or decreasing their swimming speed and efficiency,
modifying their propulsion kinematics and swimming gaits, and experiencing
different hydrodynamic interactions with their surroundings. In this article,
we review the fundamental physics of locomotion of biological and synthetic
microswimmers in complex viscoelastic fluids. Starting from a continuum
framework, we describe the main theoretical approaches developed to model
microswimming in viscoelastic fluids, which typically rely on asymptotically
small dimensionless parameters. We then summarise recent progress on the
mobility of single cells propelled by cilia, waving flagella and rotating
helical flagella in unbounded viscoelastic fluids. We next briefly discuss the
impact of other physical factors, including the micro-scale heterogeneity of
complex biological fluids, the role of Brownian fluctuations of the
microswimmers, the effect of polymer entanglement and the influence of
shear-thinning viscosity. In particular, for solutions of long polymer chains
whose sizes are comparable to the radius of flagella, continuum models cannot
be used and instead Brownian Dynamics for the polymers can predict the swimming
dynamics. Finally, we discuss the effect of viscoelasticity on the dynamics of
microswimmers in the presence of surfaces or external flows and its impact on
collective cellular behavior.
