Geodynamics is the application of the basic principles of physics, chemistry and mathematics to understanding how the internal activity of the Earth results in all the geological phenomena and structures apparent at the surface, including seafloor speading and continental drift, mountain building, volcanoes, earthquakes, sedimentary basins, faulting, folding, and more. Computational geodynamics deals with the numerical modelling of these phenomena. With the advent of modern computers, it has now become a very prolific field of research and it is one more weapon in the geo-arsenal: it allows us to test many hypothesis pertaining to the structure of the Earth, its rheology, its long-term evolution, from crustal deformation to deep mantle processes, thereby coupling different time- and length-scales. In my presentation, I will first present, in a simple way, the mathematical and computational foundations of Computational Geodynamics, and will then proceed to showcase our most recent work. I will then finish by quickly discussing the future of this rather new field of science.