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The foundations and characteristics of models and methods used in diffusion magnetic resonance imaging, with particular reference to in vivo brain imaging, are reviewed.

The first section introduces Fick's laws, propagators, and the relationship between tissue microstructure and the statistical properties of diffusion of water molecules. The second section introduces the diffusion‐weighted signal in terms of diffusion of magnetization (Bloch–Torrey equation) and of spin‐bearing particles (cumulant expansion). The third section is dedicated to the rank‐2 tensor model, the bb‐matrix, and the derivation of indexes of anisotropy and shape. The fourth section introduces diffusion in multiple compartments: Gaussian mixture models, relationship between fiber layout, displacement probability and diffusivity, and effect of the b‐value. The fifth section is devoted to higher‐order generalizations of the tensor model: singular …