Heterogeneous solid propellants are widely used in the rocket industry. They consist of oxidizer particles embedded in a polymeric binder. The combustion and ignition of such propellants remain poorly understood, partly because they are complex and driven by physical mechanisms occurring at the micrometer scale. This paper gives an overview of some recent achievements in understanding the physics of propellant burning using mesoscale direct simulations. In this modeling, the propellant microstructure is explicitly modeled, and basic governing equations (reactive Navier-Stokes equations) are solved in three dimensions at the subparticle level. Most present findings suggest a salient role of propellant microstructure. This is the case for steady combustion where the orientation of particles (assumed as spheroids) has a strong impact on burn rate, while particle shape has a more limited effect. Ignition can also be addressed; it is found that ignition physics is strongly related to particles on the surface, like flame spreading from ignited particles. Mesoscale simulations can be part of a more general multiscale strategy applied to real motors. Two applications are discussed: the first is related to the effects of the grain manufacturing process on burning rate, and the second focuses on how local small-scale burning fluctuations above the propellant surface can trigger flow instabilities in a motor. This strengthens the interest of mesoscale modeling, both for understanding physics and predicting actual systems.