A criterion for the ignition of granular explosives (GXs) and polymer-bonded explosives (PBXs) under shock and non-shock loading is developed. The formulation is based on integration of a quantification of the distributions of the sizes and locations of hotspots in loading events using a cohesive finite element method (CFEM) developed recently and the characterization by Tarver et al. [C. M. Tarver et al., "Critical conditions for impact- and shock-induced hot spots in solid explosives," J. Phys. Chem. 100, 5794–5799 (1996)] of the critical size-temperature threshold of hotspots required for chemical ignition of solid explosives. The criterion, along with the CFEM capability to quantify the thermal-mechanical behavior of GXs and PBXs, allows the critical impact velocity for ignition, time to ignition, and critical input energy at ignition to be determined as functions of material composition, microstructure, and loading conditions. The applicability of the relation between the critical input energy (E) and impact velocity of James [H. R. James, "An extension to the critical energy criterion used to predict shock initiation thresholds," Propellants, Explos., Pyrotech. 21, 8–13 (1996)] for shock loading is examined, leading to a modified interpretation, which is sensitive to microstructure and loading condition. As an application, numerical studies are undertaken to evaluate the ignition threshold of granular high melting point eXplosive, octahydro-1,3,5,7-tetranitro-1,2,3,5-tetrazocine (HMX) and HMX/Estane PBX under loading with impact velocities up to 350 ms−1 and strain rates up to 105 s−1. Results show that, for the GX, the time to criticality (tc) is strongly influenced by initial porosity, but is insensitive to grain size. Analyses also lead to a quantification of the differences between the responses of the GXs and PBXs in terms of critical impact velocity for ignition, time to ignition, and critical input energy at ignition. Since the framework permits explicit tracking of the influences of microstructure, loading, and mechanical constraints, the calculations also show the effects of stress wave reflection and confinement condition on the ignition behaviors of GXs and PBXs.