This review summarizes the work performed during the last 40 years in the field of diffusion measurement by nuclear magnetic resonance (NMR), with emphasis on biomedical diffusion imaging. Measuring molecular displacements in biological tissues in vivo has an enormous potential, but remains technically challenging. After a review of the nature of the diffusion process, the basic principles of diffusion measurements with NMR are introduced, followed by a presentation of various diffusion imaging methods. The paper covers many previously resolved theoretical and technical issues and new problems that are more specific to clinical diffusion imaging, such as the calculation of diffusion effects in the presence of multiple magnetic field gradient pulses, the elimination of motion artifacts, and the meaning of anisotropic or restricted diffusion in relation to tissue microdynamics and microstructure. The concept of diffusion imaging is then extended to blood microcirculation imaging. Finally, the current and potential clinical applications of these techniques are described.