Diffusion MRI is today used in clinical routine for detecting stroke and grading prostate tumors, as well as in clinical research studies of for instance neurological diseases and normal brain development. The overwhelming majority of the diffusion MRI measurements are performed with motion encoding by the most basic form of the pulsed-gradient spin echo sequence from the mid-60s, which is sensitive to local diffusivities, restrictions, anisotropy, flow, and exchange. While it may be convenient to have a single experiment to detect a wide range of different diffusion properties, the lack of selectivity becomes a nuisance when attempting to assign the experimental observations for a complex, heterogeneous, and anisotropic material like the living brain to a specific diffusion mechanism. This lecture will give an overview of our recent work in redesigning diffusion MRI using principles that are well known in multidimensional solid-state NMR spectroscopy and low-field NMR of porous materials. The key features of this new “multidimensional diffusion MRI” approach are gradient waveforms targeting specific motion mechanisms and multidimensional acquisition and analysis protocols wherein the different mechanisms are separated and correlated. The gradient waveforms are often borrowed from classical sample reorientation techniques such as magic-angle spinning, variable-angle spinning, and double rotation. Data inversion is performed with algorithms from multidimensional Laplace NMR, in particular the more sophisticated Monte Carlo inversion generating ensembles of plausible distributions and estimates of the uncertainties of the obtained distributions and scalar parameters. Clinical application examples include studies of microstructure in meningioma and glioma brain tumors as well as white matter degeneration in multiple sclerosis.