Geophysics talks are full of exciting colour figures showing the interior of the Earth. But are these pictures real? At best, they are only a simplified mathematical parameterisation of the true Earth; at worst they can be misleading or plain wrong. This module provides the tools to construct such models by mathematical modelling of geophysical observations, but perhaps even more importantly, shows how such models can be interpreted, and provides understanding of their limitations. Mathematical foundations are given with sections on matrix analysis, optimisation theory and statistics, with application to geophysical problems. Error estimation is considered in detail, in particular the reasons why most error estimates are close to worthless! Detailed examples are presented from all areas of geophysics, with a project to generate a model of the magnetic field of the planet Neptune. Examples also extend to modern developments, including links to Big Data and Machine Learning.