The onset of earthquakes along crustal faults, the triggering of slab avalanches or the break of a wine glass are all serious problems driven by the propagation of a dynamic rupture front. Our physical understanding of this ubiquitous phenomenon is however still far from being well-established. While Linear Elastic Fracture Mechanics (LEFM) became a successful theoretical framework to describe the stability and the slow growth of flaws within materials, it still fails at describing the dynamics of fast and sudden rupture fronts. Several experimental studies recorded the propagation of dynamic crack fronts and revealed the origin of this discrepancy. As the rupture speed approaches the one of the elastic waves, the crack stops being the simple planar object pictured by the theory. Indeed, dynamic instabilities start interfering with the crack propagation, while the fracture surface roughens and reveals the permanent interplay between the rupture front and the heterogeneities emerging from the material microstructure. In this context, the objective of the present Ph.D. work is precisely to bring novel insights in these complex and unstable dynamics using the great potential of modern computational methods. The originality of its approach consists in looking at the scale of the intimate interaction existing between the rupture front and material heterogeneities, where the crack tip spreads over some distance called the fracture process zone. To this aim, it relies on two "homemade" implementations of the elastodynamic equations running on modern computing cluster facilities. This Ph.D. work consists mainly in two parts. The first one presents a fundamental study of the interplay between a crack front and microscopic heterogeneities. This work reveals the direct impact of the heterogeneities on the local speed of the rupture front, which can even exceed the admissible values predicted for homogeneous conditions. The simulations further allow to connect the progressive roughening of the fracture surface to the "relativistic" contraction of the process zone observed when the crack speed approaches the speed of elastic waves. In the second part of the manuscript, the same formalism is applied to the study of frictional interfaces, for which the microscopic heterogeneities correspond to the scattered topography emerging from the contact between two rough surfaces. This study notably proposes a new estimation of the part of fracture energy entering the seismic energy budget. In spite of being a rather fundamental study of heterogeneous crack dynamics, this Ph.D. work finds direct implications in a large range of domains, from earthquake science to the design of more resilient materials.