The structure of a porous material is imprinted on the pattern of translational diffusion of the liquids located in the pore space. Non-invasive NMR/MRI measurements of self-diffusion are thus an attractive means of studying the microscopic structure of non-transparent and sensitive materials such as the living human brain, e.g. to detect pathological conditions or to study normal brain development. Conventional methods give information that is averaged over the entire investigated sample or volume element and fails to distinguish between properties such as pore size, shape, and orientation in case the material is heterogeneous. This chapter reviews recent progress in the field of advanced diffusion NMR/MRI methods to disentangle the confounding effects of the various pore space parameters, thereby enabling detailed characterization of anisotropic materials with complex architecture.