We report the development of laboratory based hyperspectral X-ray computed tomography which allows the internal elemental chemistry of an object to be reconstructed and visualised in three dimensions. The method employs a spectroscopic X-ray imaging detector with sufficient energy resolution to distinguish individual elemental absorption edges. Elemental distributions can then be made by K-edge subtraction, or alternatively by voxel-wise spectral fitting to give relative atomic concentrations. We demonstrate its application to two material systems: studying the distribution of catalyst material on porous substrates for industrial scale chemical processing; and mapping of minerals and inclusion phases inside a mineralised ore sample. The method makes use of a standard laboratory X-ray source with measurement times similar to that required for conventional computed tomography.