Post-seismic deformation and glacial isostatic adjustment are two processes by which the Earth deforms viscoelastically. In both cases, the details of the deformation depend on the rheological structure of the Earth as well as the forcing, which is the earthquake and further movement on the fault in the case of post-seismic deformation, and the change in load on the surface of the Earth due to the redistribution of water and ice mass in the case of glacial isostatic adjustment. It is therefore possible to learn about the Earth’s rheological structure and the processes’ respective forcings from measurements of the deformation. In order to use measurements in this way, it is first necessary to have a method of forward modelling the processes, that is, calculating the deformation due to a given forcing and in an earth model with a given structure. Given this, a way of calculating derivatives of measurements of the deformation with respect to the parameters of interest is then desirable. In this dissertation, the adjoint method is used. This, for the first time, enables efficient calculation of continuous derivatives, which have many potential applications. Firstly, they can be used within a gradient-based optimisation method to find a model which minimises some data misfit function. The derivatives can also be used to quantify the uncertainty in such a model and hence to provide understanding of which parts of the model are well constrained. Finally, they enable construction of measurements which provide sensitivity to a particular part of the model space. In this dissertation, new methods for forward modelling both post-seismic deformation and glacial isostatic adjustment are presented. The adjoint method is also applied to both problems. Numerical examples are presented in spherically symmetric earth models and, in the case of glacial isostatic adjustment, models with laterally varying rheological structure. Such examples are used to illustrate the potential applications of the developments made within this dissertation.