This thesis presents research towards the realisation of new eddy current-based multi-frequency techniques for non-destructive testing of the graphite bricks within the core of an advanced gas-cooled nuclear reactor (AGR). The graphite core suffers from two main ageing mechanisms (density reduction and cracking) and both were investigated in this thesis. There is a close relationship between the density of graphite and its electrical conductivity, which allows density changes to be monitored using eddy current techniques. A novel eddy current sensor configuration was specifically optimised for inspection of the graphite bricks, which contain the AGR fuel elements. The new sensor was designed to operate within the fuel channel bricks and its performance was validated using both Finite Element modelling and experimental studies. Results show that the new sensor offers an approximately 43 % improvement in sensitivity with depth, compared with the existing eddy current sensor currently used for routine reactor core inspections. The thesis also considers the novel solutions to the problem of determining the depth profile of the electrical conductivity of the bricks. The depth profile of electrical conductivity could allow the density profile to be inferred. This study has focused on the formulation of a suitable non-linear inversion algorithm that can accurately estimate the depth profile of the electrical conductivity based on the measurements collected from a real reactor core, through the implementation of a new forward model calibration and constraining techniques. Two different Tikhonov based inversion algorithms have been studied, namely regularised Gauss Newton and regularised Levenberg Marquardt. The former algorithm was already implemented for this application in a previous PhD project. This research has now extended the algorithm with new constraining techniques to improve its performance. The results from the reconstructed profiles using a data from a representative laboratory sample show that the graphite cross-sectional profile can be reproduced with 98.7 % accuracy when the new constraining technique is applied within the algorithm. A new application of the regularised Levenberg Marquardt algorithm was adapted for the conductivity profiling as part of this work, and the performance of this algorithm was compared with the regularised Gauss Newton method. The two algorithms were compared based on the reactor data inverse solutions. Slightly faster convergence rate was observed from the regularised Gauss Newton algorithm, but it suffers from inaccuracies with increasing depth from the graphite bore. The regularised Levenberg Marquardt algorithm tends to produce much more accurate profiles, although it takes longer to arrive at the final solution. Comparisons between the reconstructed profiles of the fuel channel bricks using RLM and the measurements from the trepanned sample also showed reasonable agreements between one another, with mean profile errors ranging between 1.5 % and 16.9 %. This is the first time that such good agreement has been obtained from reactor data. Experimental studies concerning with realistic sub-surface crack have been carried out using the new sensor. During these studies an attempt was made to extend the existing method of determining crack location. The results from these studies show that the new sensor along with the extended data processing method allows a detection of realistic crack that was 34 % of the entire graphite wall, whereas the existing sensor was only detected a crack that is 48 % of the entire graphite wall.