The analysis of masonry arch bridges is still a challenge for engineers due to its complex and nonlinear behavior. In practice, structural behavior of masonry arch bridges is studied by following relatively simple methods, e.g. limit analysis, which does not require a significant number of parameters. Two-dimensional nonlinear finite element models are also common in the literature; however, these do not reflect the full structural response, since they neglect the out-of-plane actions. These models neglect spandrel walls, 3D point load effect and skew arches, among other effects. The objective of this study is to present a methodology that can simulate three-dimensional masonry arch bridge behavior comprehensively and can include various possible failure mechanisms. Discrete element method (DEM), which is a discontinuum approach, is used to understand the influence of essential structural components, such as the arch barrel, spandrel wall and back-fill material on several masonry arch structures. The masonry units are modeled using discrete blocks and back-fill material is generated as a continuum mesh, based on the plasticity theory. Load carrying capacity and related collapse mechanisms are investigated through a set of parametric studies on the mechanical properties of back-fill material. Out-of-plane spandrel wall failures were further explored by taking advantage of a discontinuous approach. The results indicated that soil characteristics (elastic modulus, internal friction angle and cohesion) have remarkable influence on the behavior and load carrying capacity of the masonry arch bridges. Further, the analyses are also validated with previously published experimental work as well as an existing historical bridge.