A knowledge of stereology (i.e. proper sampling), the opportunities provided by computers for image analysis (i.e. image segmentation, image registration, data base exploration, 3D reconstruction), and the strengths (i.e. non-invasive) and limitations (i.e. finite resolution, image artefacts) of medical imaging equipment must all be combined for reliable quantitative magnetic resonance imaging (MRI), the goal of which is to obtain a deeper understanding of the structure, function, life cycle and evolution of the human body, especially the brain, and a more objective diagnosis of disease and assessment of its response to treatment. In this article we illustrate the first of these requirements. We describe the application of proper sampling strategies and efficient computer-based counting procedures for obtaining unbiased estimates of volume by the Cavalieri method and of surface area from vertical sections. In particular, we estimate the volume of a brain tumour from Cavalieri sections, the volume of grey matter in the cerebral hemispheres from Cavalieri slices and the surface area of the cerebral cortex from vertical sections. The estimates obtained are mathematically unbiased. In each case, we assess the precision of the estimates empirically. Application of formulae available for predicting the precision of volume estimates obtained using the Cavalieri sections and slices methods is also described.