The course is an exploration of the recent theoretical, experimental and numerical advancements in the modelling of non-Brownian granular suspensions. The quest is mainly motivated by the growing scientific and engineering interest in geophysical flows as a consequence of the climate change and the severe impact of hydro-geological catastrophic events on socio-economic activities. Among geophysical phenomena, the course focuses on water-saturated sub-aerial and sub-aqueous debris flows, hyper-concentrated flows, underwater turbidity currents, creeping and fluid-like movements of soil in landslides as well as on the sediment transport in rivers and along lake and ocean shores, both at the bed and in suspension. Indeed, the presence of a liquid, which saturates the interstices between grains, formidably expands the parameter space of granular flows, the mixture exhibiting behaviours typical of either viscous shear-thinning fluids or dense granular flows depending essentially on the average distance between solid particles (i.e. the particle concentration), the relative velocity of particles and the electro-chemo-mechanical properties of the mixture components. The scope of the course is the investigation of the purely hydraulic problem, thus the liquid is water while particles are in general coarse and inert (non-colloidal and non-adhesive) and, therefore, only their mechanical properties are considered. Nonetheless, a brief excursus on the rheology of more complex suspensions is also planned for the sake of completeness. The opening of the course is devoted to highlight the striking differences between (dry) granular flows and granular suspensions. Granular suspensions are first classified on the basis of the relative velocity of particles, namely of the particle Reynolds number, into “viscous” and “inertial”, which determines the nature of dominant fluid-solid interactions. Then, “dilute”, “semi-dilute” and “dense viscous suspensions” are distinguished which differ in the number concentration of particles and, therefore, on the role of inter-particle contacts. The dynamics of the granular suspensions under different boundary/initial conditions as well as different driving forces are investigated and modelled highlighting the most recent advancements in the subject. Continuum and discrete approaches are considered. The continuum approach comprises a single-phase, also referred to as “single effective fluid”, or two-phases which indicates that the “theory of mixture” is adopted. As for the discrete approach, it necessarily requires the use of numerical methods to solve the fluid-particle-coupled continuity and momentum equations. The coupling and the inter-particle contacts can be obtained with a point-particle approach or, for “large particles”, by fully resolving the flow field around the particles. The effects due to the presence of turbulent vortices, to the poly-dispersion of particle size and to the particle shape are also considered. Finally, models are described and applied to the aforementioned geophysical flows.