Abstract GPUSPH is a fully three-dimensional model for the simulation of the thermal and rheological evolution of lava flows that relies on the Smoothed Particle Hydrodynamics (SPH) numerical method. Thanks to the Lagrangian, meshless nature of SPH, the model incorporates a more complete physical description of the emplacement process and rheology of lava that considers the free surface, the irregular boundaries represented by the topography, the solidification fronts and the non-Newtonian rheology with temperature-dependent parameters. GPUSPH follows the very general Herschel–Bulkley rheological model, which encompasses Newtonian, power-law and Bingham flow behaviours, with both constant and temperature-dependent parameters, and can thus be used to explore in detail the impact of rheology on the behaviour of lava flows and on their emplacement. To illustrate this possibility, we present some preliminary applications of the model for studying the rheology of lava flows with different constitutive relationships and thermal regimes using the real topography of the Mt Etna volcano.