Glass fibre composites are common materials used in high voltage applications as insulating materials that provide good structural integrity. The aim of this thesis is to develop a method of studying the failure in such materials by measuring the dielectric strength on micro- and meso- scale samples, consisting of single fibre filaments and fibre bundles respectively embedded in epoxy resin. To do this, a body of relevant knowledge has been amassed, which is complemented with finite element analysis giving detailed insight into the electric field distribution in the microstructure of fibre composites. A method of producing virtually defect free single fibre samples has been developed where a filament is hung down tubes and cast in epoxy resin. A similar method was developed for producing bundle samples, however this needs some slight correction in order to prevent exothermic reactions. The dielectric strength of these samples are measured by applying a continuously increasing voltage until discharge is recorded. To evaluate the method micro- and meso- scale samples were prepared of three different fibres and their dielectric strengths measured. This evaluation showed that the method can be used to measure a definitive lower bound in the dielectric strength of fibre composites. However, the method can not definitively determine the location of the discharge, which is necessary to verify conclusions about the materials properties. To progress the method, the dielectric strength of neat epoxy samples of the same dimensions as the fibre composite samples should be investigated. Increasing the tolerance of the measurement setup should also be investigated as this could help by increasing the power of the discharge leading to more severe damage in the material.