A knowledge of stereology (i.e. proper sampling), the opportunities provided by computers for image analysis (i.e. image segmentation, image registration, database exploration, 3D reconstruction), the strengths (i.e. non-invasive) and limitations (i.e. finite resolution, image artefacts) of medical equipment and an understanding of the interface between machine and a living (i.e. moving) subject 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 and especially the brain, and a more objective diagnosis of disease and assessment of its response to treatment. This thesis is concerned with the first of these requirements and describes the development of an interactive software package, EASY MEASURE, which enables convenient computer-based application of stereological methods for estimating volume, surface area and length. In particular, the lengths of four vertebral columns are estimated from single composite mid-line sagittal MR images, the surface areas and volumes of five pears are estimated from vertical sections, and the mean thickness of the cerebral cortex in six formalin-fixed brains and one paradigm in-vivo MR brain dataset is estimated from vertical sections. The estimates obtained are mathematically unbiased. In each case, the precision of the estimates is investigated empirically. The application of formulae for predicting the precision is also described.