Biological microorganisms swim in different types of fluids using a range of diverse motions and are often found in complex geometries. Their study is a rich field full of outstanding problems. One aspect that has attracted a lot of attention is the role played by hydrodynamic interactions, including those between a cell and its fluid environment (fluid-cell), or between neighbouring cells (cell-cell). These interactions can affect dramatically the dynamics of these swimmers.

This dissertation investigates the dynamics of filaments interacting hydrodynamically with a fluid through slip boundary conditions or with other filaments and is composed of two separate parts. The first part of the thesis focuses on a single filament characterised by a slip boundary condition, which is a property displayed by many non-Newtonian fluids. I propose a waving sheet and a waving cylinder model to demonstrate a possible enhancement of swimming by such slip effect. The results are in good agreement with previous experimental and numerical studies. In subsequent work, I extend the classical slender-body theory to replace the no-slip boundary condition by a finite slip length.

The second part of the thesis addresses the nature of hydrodynamic interactions between filaments - a phenomenon that occurs widely in the biological world. By developing a new method for integrating hydrodynamic singularities between interacting filaments, I show how the force on the filament can be evaluated analytically. Using this result, I study the specific problem of bacterial flagellar bundling. This complex process is studied in two steps. Firstly, using a simpler geometry, I propose a model with elastic filaments to reveal the dynamics of bundling and unbundling. I then expand upon this to consider the full helical geometry of a bacterial flagellum, and develop a theoretical model for the pathway to synchronization. In each case, either by considering simple geometries or through the use of asymptotic methods, I capture in the model the main physical features and compare these to previous experimental and numerical results, thereby making important progress toward our understanding of the physics of flagellar bundling.