Failure of materials and interfaces is mediated by the propagation of cracks. They nucleate locally and slowly then, as they exceed a critical size, accelerate and reach speeds approaching the speed of sound of the surrounding material. As they propagate, they dissipate energy within a confined region at the crack tip, which approaches a mathematical singularity. As a result, the initiation and propagation of cracks is a spatial and temporal multiscale phenomenon. The framework of linear elastic fracture mechanics captures many aspects related to the dynamic propagation of cracks in homogeneous media. However, the propagation of a crack within a medium with heterogeneous elastic or fracture properties cannot be addressed theoretically. It is in these complex, heterogeneous cases that numerical simulations and experiments shine. The material heterogeneity introduces additional length scales to the problem, which characterize the geometrical properties or spatial correlation of the heterogeneities. The interaction of these geometrical length scales with fracture mechanics related ones is not well understood, but it could provide crucial insights for the design of new materials and interfaces with unprecedented fracture properties. This thesis investigates different aspects of crack nucleation and propagation in heterogeneous materials and interfaces, including nucleation of mode II ruptures on interfaces with random local properties, dynamic mode II rupture propagation within elastically heterogeneous media, and dynamic mode I rupture propagation within a material with periodic heterogeneous fracture energy. In this context, when considering mode II dynamic fracture problems, we are making an analogy to frictional interfaces. In fact, the onset of frictional motion is mediated by crack-like ruptures that nucleate locally and propagate dynamically along the frictional interface. To investigate the complex interaction between fracture mechanics and geometry related length scales we adopt a combined approach using numerical, theoretical, and experimental methods. The numerical simulations consider a continuum governed by the elastodynamic wave equation and allow for a displacement discontinuity (the rupture) along a predefined interface. Depending on the nature of the heterogeneity, the fracture propagation problem is solved using either the finite-element or the spectral-boundary-integral method. Here, we introduce a novel three-dimensional hybrid method, which combines the two former numerical methods to achieve superior computational performance, while allowing modeling of local complexity and heterogeneity. From the experimental side we use state-of-the-art techniques, including ultra-high-speed photography, digital image correlation, and multi-material additive manufactured polymers. We show that random local strength results in three different nucleation regimes depending on the ratio of correlation length to critical nucleation size. We show that elastic heterogeneity parallel to the fracture interface promotes transition to intersonic crack propagation in mode II cracks by means of reflected elastic waves. Finally, our experimental results of a crack propagating within a material with heterogeneous fracture energy show that the crack abruptly adjusts its speed as it enters a tougher region and allow us to derive an equation of motion of a crack at a material discontinuity.