The electromagnetic induction problem is non-linear, and thus is very difficult to solve for all but the simplest symmetries. Because of this, quantitative modelling of the conductivity structure from geomagnetic depth sounding data is expensive and time consuming, and the possibility that the anomaly is produced by channelling of regionally induced currents may invalidate the results. For this reason traditional methods of analysis are generally qualitative in nature, with quantitative information estimated on the basis of simplified models of the anomaly. The theory and assumptions used in these traditional methods are studied in this thesis, and the range of their applicability is investigated. To avoid the current channelling complication, and to also get a linear relation between the model and the data, the problem is reformulated with the subsurface current density as the model parameter, rather than the conductivity. The disadvantage of this formulation is that models that fit the data are very non-unique. The character of this non-uniqueness has been explored using Backus-Gilbert appraisal, and by the construction of unconstrained models. The results indicate that reasonable resolution of the true model's horizontal features is possible, but that vertical resolution will be lacking. To counter this, the infinite range of possible models is constrained by introducing expected physical features of the true model into the model construction algorithm.' This construction algorithm was tested using data generated from a variety of artificial models, and was successful in resolving both the horizontal and vertical positions of the major features in all of them. The algorithm was then used to determine the subsurface current structure for real data taken across the Cascade anomaly in Washington State.