The focus of this thesis is on methods for simulating microscopic objects in viscous fluids. The long range hydrodynamic interactions between submerged objects couple their motions and establish a computationally challenging problem. Depending on the size of these immersed objects, thermal fluctuations of the fluid may induce random Brownian motion or the motion may be deterministic. This thesis is concerned with both cases.

In the deterministic setting, a methodology for modelling and simulating inextensible, flexible filaments is introduced. Specialised parameterisations are used to describe the filament state and the method admits any hydrodynamic model for the filament-fluid interactions which is compatible with the inertialess Stokes regime. A computationally-scalable implementation is discussed which employs implicit time integration and enforces constraints using either geometric integration or Lagrange multipliers. The potential of the method is explored by applying it to the study of tethered filaments. New filament motions and collective behaviours are observed and discussed.

The Brownian motion of rigid bodies is considered in the stochastic case. An existing method for the positions of spheres is generalised to the positions and orientations of arbitrarily-shaped rigid particles. Geometric integration is used for the orientations and the appropriate stochastic equation is established. The method is applied to problems inspired by liquid crystals to demonstrate that it produces the correct statistics and to suspension rheology to showcase its capabilities.