This thesis is devoted to the theoretical description of two experimentally observed phenomena which occur at small scales. We first address the synchronization of swimming microorganisms. Motile microorganisms swim in a fluid regime where inertia is unimportant and viscous stresses dominate. In this limit the flow field due to a swimmer affects the motility of nearby cells, a fact which is biologically important as microorganisms such as spermatozoa are often found in high-density suspensions. A particular consequence of these fluid-based interactions is the synchronization of the flagella of some microorganisms, and in particular spermatozoa, observed to occur when these cells are swimming in close proximity. Using theoretical analysis it is demonstrated that two infinite sheets passing waves of a prescribed shape, will not synchronize in a Newtonian fluid if the shape of the waveforms has sufficient symmetry because of the kinematic reversibility of the Stokes equations. The sinusoidal waveforms of Taylor's swimming sheet fall into this category, and will thus not dynamically synchronize in a Newtonian fluid. It has been observed that excess symmetry similarly curbs synchronization in other models. For a sinusoidal sheet, a geometric perturbation must therefore be added to break the necessary front/back symmetry, and give rise to a time-evolution of phase toward the synchronized state. Alternatively, instead of a geometric symmetry-breaking, it is also shown that synchronization can occur if the kinematic reversibility of the field equations is removed, as is the case for a viscoelastic fluid. In such a scenario the phase always evolves to a stable in-phase conformation where the energy dissipated by the swimmers is minimized. Finally it is shown that finite size effects act to bring swimmers closer together and then we show in this regime that elastic deformations caused by fluid structure interactions play a dominant role in synchronization dynamics. Additionally, motivated by recent experiments, we consider theoretically the compression of droplets pinned at the bottom on a surface of finite area. We show that if the droplet is sufficiently compressed at the top by a surface, it will always develop a shape instability at a critical compression. When the top surface is flat, the shape instability occurs precisely when the apparent contact angle of the droplet at the pinned surface is pi, regardless of the contact angle of the upper surface, reminiscent of past work on liquid bridges and sessile droplets as first observed by Plateau. Past the critical compression, the droplet transitions from a symmetric to an asymmetric shape. The force required to deform the droplet peaks at the critical point then progressively decreases indicative of catastrophic buckling. We characterize the transition in droplet shape using illustrative examples in two dimensions followed by perturbative analysis as well as numerical simulation in three dimensions. When the upper surface is not flat, the simple apparent contact angle criterion no longer holds, and a detailed stability analysis is carried out to predict the critical compression
