Innate immunity is crucial in the early stages of resistance to novel viral infection. The family of cytokines known as the interferons (IFNs) form an essential component of this system : they are responsible for signalling that an infection is underway and for promoting an antiviral response in susceptible cells. Despite the importance of the IFNs in viral infections, many questions about the dynamics of IFN production and activity remain unanswered. Addressing these questions forms the first half of this thesis. lt begins with an introduction to innate immunity, and a review of existing research . Focus then falls upon a simple experimental system : the in vitro infection of cell monolayers by Herpes simplex virus and the resulting IFN response. A stochastic spatial model of viral infection and consequent IFN production and activity is constructed. Using this model, simulations of infections under varying initial conditions suggest the existence of critical doses, at which the qualitative behaviour of infection changes. Implications for IFN activity in in vivo infections are highlighted, as well as potential applications of the model, particularly in within-host modelling. The data used to parameterize this model come from widely used experiments called plaque assays: infection spreads in cell monolayers in vitro, leaving regions of dead cells, known as plaques. The second half of the thesis considers the dynamics of plaque formation and, in particular, the phenomenon of cometing, where in plaques unexpectedly streak across monolayers forming patterns that resemble comets. Several theories behind comet formation have been proposed in the literature, though the underlying mechanism is not understood . A detailed investigation is carried out here: previously voiced hypotheses are tested, and a method for controlling comet formation is developed; cometing is found to be a purely fluid dynamic phenomenon. The thesis concludes with an overview of the results obtained , and a discussion of potential applications and future directions for research.