Understanding how things break and slide is of paramount importance to describe the dynamics of a broad range of physical systems. This includes day-to-day problems such as the breaking of a glass of wine or the sliding of skis on snow, but also engineering systems with, for example, the braking of a car or the failure of a structural component, up to geophysics and earthquake science. These two topics, fracture, and friction, seem unrelated at first but share similar physical characteristics: they are mediated by the propagation of rupture fronts. In both cases, a ruptured state (a crack, or a slipping patch) invades an intact state (the unbroken material, or a sticking interface). While the questions related to fracture and friction are ubiquitous, our physical understanding of these phenomena is far from complete. The Linear Elastic Fracture Mechanics framework accurately describes the stability of defects in materials and their slow growth but fails to explain the unstable three-dimensional dynamics at play in rapid fracture. Concepts from fracture mechanics have been successfully applied to the propagation of frictional rupture fronts, but fundamental differences remain due to the complex behavior of the friction coefficient itself, being dependent on the slipping rate and the state of the microcontacts at the interface between two solids. Hence, dynamic rupture exhibits a richness of behaviors. Amongst other things, the interaction between a front and material heterogeneities, boundary conditions, and finite geometry can significantly alter the dynamics of a rupture. This work aims at exploring this richness in dynamic rupture, taking advantage of efficient computational methods that solve the elastodynamic equations. The use of modern computing methods allows modelling ruptures down to the small dissipation length scale near the tip of a rupture, the process zone size. The two software used in this work are open-source codes that were developed in the Computational Solid Mechanics Laboratory at EPFL, Switzerland, where this thesis was conducted.