The aim of the present thesis is to model numerically the dynamic failure behavior of concrete at the mesoscale. Concrete is the world's most widely used construction material and the structures made of it are subjected to dynamic loads during their long service lives. However, the physical mechanisms leading to its dynamic failure are not well understood. We consider concrete from a mesoscopic point of view, where coarse aggregates, mortar paste and the interfacial transition zone can explicitly be represented. The explicit modeling of the first-level heterogeneities makes mesoscale the most convenient scale to study fracture. The well-established finite-element method is used for the numerical analysis. Fracture is modeled by cohesive elements, which are inserted dynamically when the stress at an inter-element boundary exceeds a local threshold. A robust and scalable parallel implementation of the method is utilized to analyze large scale three-dimensional models. We begin analyzing the influence of material heterogeneities by modeling limited aggregate patterns. This approach allows us to focus on a single propagating crack and to isolate the changes in the elastic field and stress wave interactions. Fracture behavior is analyzed by using the Brazilian splitting test with a physically identified heterogeneity and the pre-strained plate configuration. The crack tip is shown to be repelled by the stiffer inclusions while it is attracted to the denser ones. Furthermore, by introducing denser media around the crack propagation direction, we are able to decrease its steady-state velocity. The crack is slowed down by trapping the mechanical waves in the denser region thus limiting the energy flowing to its tip. We argue that the criterion for crack branching should be based on the available energy instead of the crack tip velocity as suggested by early works. A three-dimensional mesoscale model is then developed to obtain the total mechanical response of concrete. The specimens are subjected to uniaxial tensile loading in a dynamic setting and the so-called ``rate effect'' is investigated. Micro-inertial forces are shown to be insufficient in two and three dimensions making a rate-dependent material model necessary to account the strengthening due to the moisture content in pores. Finally, a percolation analysis is carried out to investigate cluster properties such as number and size distribution. With a parametric study, we consider physically admissible ranges of coarse aggregate contents, random material parameters, applied strain rates and simulation box sizes. The size distribution of the clusters is defined by a power-law fit regardless of the parameters mentioned above, indicating a universal phenomenon is at play. Damage evolution is further studied by comparing the percolated damage clusters with the final cracking map.