In this paper, we review the state of the art in diffusion magnetic resonance imaging (dMRI) and we present current trends in modelling the brainʼs tissue microstructure and the human connectome. dMRI is today the only tool that can probe the brainʼs axonal architecture in vivo and non-invasively, and has grown in leaps and bounds in the last two decades since its conception. A plethora of models with increasing complexity and better accuracy have been proposed to characterise the integrity of the cerebral tissue, to understand its microstructure and to infer its connectivity. Here, we discuss a wide range of the most popular, important and well- established local microstructure models and biomarkers that have been proposed from these models. Finally, we briefly present the state of the art in tractography techniques that allow us to understand the architecture of the brainʼs connectivity.