This work presents results from numerical studies of dynamics in three classical non-linear field theories, each of which possesses stable, localized solutions called solitons. We are mainly interested in two of the theories, known as Skyrme models, which have had application in various areas of physics. The third, which describes the dynamics of a complex scalar field and its solitonic solutions (named Q-balls), is principally viewed as a model problem for the development of solution techniques. In all cases, complicated, time dependent, non-linear partial differential equations in several spatial dimensions must be solved, and this necessitates a computational approach. A particular focus of the work is the simulation of high-energy collisions of the solitons. The chief contributions of this work come from simulations performed within the context of a Skyrme model in two spatial dimensions. We concentrate on the rich phenomenology seen in high-energy scattering of pairs of these objects, and the outcome of head-on and off-axis collisions. The study of instabilities seen in previous simulations of Skyrme models is of central interest. Our results confirm that the governing partial differential equations become of mixed hyperbolic-elliptic type for interactions at sufficiently high-energy. We present strong evidence for the loss of energy conservation and smoothness of the dynamical fields in these instances. This supports the conclusion that the initial value problem at hand becomes ill posed, so that the observed instabilities result from the nature of the equations themselves, and are not numerical artifacts. Our calculations incorporate parallel adaptive mesh refinement, which allows us to deal efficiently with the significant dynamical range exhibited in the simulations.