Non-invasive virtual histology of white matter tissues is the ultimate promise of diffusion-weighted MRI (DW-MRI). This hope is fueled by the exquisite sensitivity of DW-MRI to micrometer-scale displacement of diffusing water molecules, averaged over a whole voxel. Combined with adequate mathematical modeling, DW-MRI therefore has the potential to provide unprecedented, microscope-like insight into the microstructure of the brain, with important consequences in brain neurosciences, neurology and psychiatry. Many closed-form analytical models have been proposed to relate the DW-MRI signal to morphological characteristics of the axons and glial cells of the white matter. These models generally make assumptions about the tissue and the diffusion processes which often depart from the biophysical reality, limiting their reliability and interpretability in practice. Monte Carlo simulations of the random walk of water molecules are widely recognized to provide near numerical groundtruth for DW-MRI signals. However, they have mostly been limited to the validation of simpler models rather than used for the estimation of microstructural properties. This thesis proposes a general framework which leverages Monte Carlo simulations for the estimation of physically interpretable microstructural parameters such as indices of axon diameter and density, both in single and in crossing fascicles of axons, with no restriction on the data acquisition protocol. Monte Carlo simulations of DW-MRI signals, or fingerprints, are pre-computed for a large collection of microstructural configurations. At every voxel, the microstructural parameters are estimated by finding a sparse optimal combination of these fingerprints. The final complexity of the model is thus solely determined by the level of tissue detail incorporated in the Monte Carlo simulations. The parameter estimation requires no meta-parameter tuning. The superposition approximation for DW-MRI signals in the presence of multiple fascicles of axons, often taken for granted by state-of-the-art models, was thoroughly verified. This enabled a dramatic reduction of the size of the dictionary and of the ensuing fingerprint matching. Our approach was then validated extensively on synthetic data as well as on a diversity of in vivo and ex vivo datasets. It was shown to systematically provide more robust and physically interpretable tissue parameters than a range of traditional closed-form models claiming similar biophysical complexity. An accelerated method based on efficient convex estimation and a deep feed-forward neural network was developed to accommodate our framework to larger-scale studies. This fast two-stage procedure reduced the execution time by several orders of magnitude while maintaining a similar level of estimation accuracy. All software tools related to this work will be shared upon publication of this thesis. They can easily be integrated with for instance, recently-released Monte Carlo simulators offering exciting new levels of tissue realism. This is intended to make microstructure imaging based on Monte Carlo simulations accessible to as broad an audience as possible, for future population studies and the general advancement of the field.