To be reliable, Earth models used for mineral exploration, or other subsurface investigations, should be consistent with all available geological and geophysical information. Geophysical inversion provides the means to integrate geological data such as lithology and structure, physical property data such as measurements taken on rock samples, and geophysical survey data. Inversion is a computational process that recovers models of the subsurface that could have given rise to measured data from a geophysical survey, for example, seismic, gravity, magnetic, electrical and electromagnetic measurements, while maintaining consistency with the geological knowledge available. We are performing geophysical inversion on unstructured meshes because they provide the flexibility required to efficiently incorporate complicated geological information. There are some significant challenges involved in working with unstructured meshes. These include the generation and storage of unstructured meshes and developing appropriate matrix operators for numerical forward modelling and inversion. A major challenge is developing appropriate smoothness regularization and gradient operators on unstructured triangular and tetrahedral meshes. We apply our methods to a 3D example inspired by a real mineral exploration scenario.