This dissertation contains original research on a range of problems involving the locomotion of different types of microswimmers, including both biological microorganisms and artificial colloids. Due to the physical constraints imposed by the inertialess hydrodynamics at these small scales, these swimmers rely on multiple locomotion strategies that are unfamiliar from the macroscale. The results presented here answer several theoretical questions concerning the fundamentals of specific propulsion mechanisms, as well as the interactions of a wide range of different microswimmers with geometrically complex environments.

For a general spherical squirmer-type microswimmer we first analyse the swimming dynamics in a periodic three-dimensional lattice of obstacles numerically, and obtain a phase diagram detailing qualitatively different kinds of trajectories. These range from nearly straight over diffusive to trapped, depending on the squirming parameter and the lattice packing density. We then explain these results theoretically using a combination of near- and far-field hydrodynamic arguments. Importantly, we predict qualitatively different dynamics of pusher'-type swimmers, such as bacteria, and puller'-type swimmers, such as algae, in a geometry that is representative for soil as a biologically relevant domain.

Next, we derive the singularity representation for the solution of the Stokes equations in the two cases of a point torque outside a rigid sphere, and a point torque outside a spherical bubble. In the axisymmetric case outside a rigid sphere, the solution takes an extremely simple form that is reminiscent of the solution for a point charge outside a grounded sphere in electrostatics, and is rationalised with a similar geometrical argument. In addition, we repeat the analysis for a point source.

We apply these results to an analysis of the swimming dynamics of a single peritrichous bacterium, specifically the physical mechanisms that lead to the formation of a flagellar bundle behind the cell during its forward motion. We categorise the forces at play into direct' and indirect' depending on whether they are due to hydrodynamic interactions between the flagellar filaments, or triggered by flows around the cell body due to its motion, and demonstrate using a minimal theoretical model that under very general conditions the latter dominate in all but the final stages of bundle formation. For parameter values that are representative for the model bacterium Escherichia Coli we perform a full dynamic elastohydrodynamic simulation to analyse the relative strength of hydrodynamic and elastic effects along the full length of the flagella during the bundling process.

On the topic of artificial microswimmers we next examine theoretically the stochastic dynamics of a self-propelled colloid that is dissolving over time, as motivated by recent experiments aimed at the design of active particles suitable for biomedical applications. We present two models that differ in the details of the dissolution mechanism, and in each case derive analytical expressions for the particle life time and mean squared displacement due to active diffusion. A new dimensionless parameter emerges, classifying trajectories into globally ballistic and globally diffusive depending on the ratio of particle life time to rotational diffusivity, and we obtain a hierarchy of our models in all regimes that quantifies the limit of control that can be exerted on the motion of dissolving colloids.

Expanding on the topic of control and manipulation on the microscale, we finally analyse the entrapment of passive cargo by a magnetically actuated spheroidal roller near a rigid wall. We predict that as such a roller propels along an interface, it is able to collect and transport passive cargo particles in its path by entrapping them inside a vortex to its side. This apparent violation of the Stokes flow reversibility is facilitated through irreversible steric interactions between the cargo and the interface, and analogous to the microfluidic technique of deterministic lateral displacement. We combine finite element simulations of the flow field due to the roller with an effective model for cargo migration to generate a phase diagram of entrapment as a function of roller aspect ratio, cargo size and cargo location, and predict that flat rollers are able to trap cargo for the largest range of parameter values.