The propulsion of low Reynolds number swimmers has been widely studied, from the swimming sheet models of Taylor (1951), which were analogous to swimming spermatozoa, to more recent studies by Smith (2010) who coupled the boundary element method and method of regularised Stokeslets to look at cilia and flagella driven flow. While the majority of studies have investigated the propulsion and hydrodynamics of spermatozoa and bacteria, very little has been researched on bi-flagellate green algae. Employing an immersed boundary algorithm and a flexible beat pattern Fauci~(1993) constructed a model of a free-swimming algal cell. However, the two-dimensional representation tended to over-estimate the swimming speed of the cell. Jones~\etal~(1994) developed a three-dimensional model for an idealised bi-flagellate to study the gyrotactic behaviour of bottom-heavy swimmers. However, the un-realistic cell geometry and use of resistive force theory only offered order of magnitude accuracy. In this thesis we, investigate the hydrodynamics of swimming bi-flagellates via the application of the method of regularised Stokeslets, and obtain improved estimates for swimming speed and behaviour. Furthermore, we consider three-dimensional models for bi-flagellate cells with realistic cell geometries and flagellar beats.We investigate the behaviour of force- and torque-free swimmers with bottom-heavy spheroidal bodies and two flagella located at the anterior end of the cell body, which beat in a breast stroke motion. The cells exhibit gravitactic and gyrotactic behaviour, which result in cells swimming upwards on average in an ambient fluid and also towards regions of locally down-welling fluid, respectively. In order to compare how important the intricacies of the flagellar beat are to a cell's swimming dynamics we consider various beat patterns taken from experimental observations of the green alga \Rein~and idealised approaches from the literature. We present models for the bi-flagellate swimmers as mobility problems, which can be solved to obtain estimates for the instantaneous translational and angular velocities of the cell. The mobility problem is formulated by coupling the method of regularised Stokeslets with the conditions that there is no-slip on the surface of the body and flagella of the cell and that there exists a balance between external and fluid forces and torques. The method of regularised Stokeslets is an approach to computing Stokes flow, where the solutions of Stokes equations are desingularised. Furthermore, by modelling the cells as self-propelled spheroids we outline an approach to estimate the mean effective behaviour of cells in shear flows.We first investigate bi-flagellate swimming in a quiescent fluid to obtain estimates for the mean swimming speed of cells, and demonstrate that results for the three-dimensional model are consistent with estimates obtained from experimental observations. Moreover, we explore the various mechanisms that cells may use to re-orientate and conclude that gyrotactic and gravitactic re-orientation is due to a combination of shape and mass asymmetry, with each being equally important and complimentary.Next, we compare the flow fields generated by our simulations with some recent experimental observations of the velocity fields generated by free-swimming \rein, highlighting that simulations capture the same characteristics of the flow found in the experimental work. We also present our own experiments for \rein~and \Dunny~detailing the trajectories and instantaneous swimming speeds for free-swimming cells, and flow fields for trapped cells. Furthermore, we construct flagellar beats based upon experimental observations of \dunny~and \textit{D. bioculata}, which have different body shapes and flagellar beats than \chlamy. We then compare the estimates for swimming speed and re-orientation time with \rein, highlighting that, in general, \Dun~achieve greater swimming speeds, but take longer to re-orientate.The behaviour of cells in a shear flow is also investigated showing that for sufficiently large shear, vorticity dominates and cells simply tumble. Moreover, we obtain estimates for the effective cell eccentricity, which, contrary to previous hypotheses, shows that cells with realistic beat patterns swim as self-propelled spheres rather than self-propelled spheroids. We also present a technique for computing the effective eccentricity that reduces computational time and storage costs, as well as being applicable to unordered image data.Finally, we examine what effects interactions with boundaries, other cells, and obstacles have on a free-swimming cell. Here, we find that there are various factors which affect a cell's swimming speed, orientation and trajectory. The most important aspect is the distance between the interacting objects, but initial orientation and the flagella beat are also important. Free-swimming cells in an unbounded fluid typically behave as force-dipoles in the far field, and we find that for cell-to-cell and cell-to-obstacle interactions the far field behaviour is similar. However, swimming in the proximity of a boundary results in the flow field decaying faster. This implies that hydrodynamic interactions close to solid no-slip surfaces will be weaker than in an infinite fluid.
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