Brain stroke is an age-related illness which has become a major issue in our ageing societies. Early diagnosis and treatment are of high importance for the full recovery of the patient, as reminded in Anglo-Saxon countries by the abbreviation FAST (Face, Arm, Speech, Time) referring to both the four major visible signs and the necessity to act fast. In this respect, Computed Tomography (CT) and Nuclear Magnetic Resonance (NMR) imaging are key diagnostic tools in clinical practice. Unfortunately, not only these modalities can neither be transported nor rapidly usable, which would allow early treatment (especially in rural environments), but also cannot be brought to the bedside of the patient to monitor the evolution of the disease. Microwave Imaging (MWI) is a potential candidate to provide fast and accurate diagnostic insights for brain stroke pathological states. The head of the patient is illuminated with low-power microwave waveforms (non-ionizing radiations), whose backscattered signals are used to generate either images of its internal structures, distributions, patterns and shapes (qualitative imaging) or directly its physical parameters such as the dielectric contrast and the permittivity values (quantitative imaging). The technology relies on the high sensitivity of microwaves on the water content of tissues to allow for the discrimination between pathological and healthy regions. This thesis focuses on both the forward modeling of the electromagnetic phenomena arising in biological tissues and the inverse scattering problem for imaging in the differential MWI (dMWI) scenario for brain stroke monitoring. It is intrinsically interdisciplinary as it requires knowledge in Biology, Medicine, Physics, Chemistry, and Engineering. In order to investigate the challenges arising in brain MWI, it is crucial to have accurate and efficient solvers to model electromagnetic (EM) fields at UHF/SHF-bands. The head is a distributed, heterogeneous, and lossy scatterer for which existing solvers are known to struggle at higher frequencies. Volume Integral Equation (VIE) formulations and MultiGrid (MG) approaches are investigated to find the actual solution of the field distributions for large scale problems. The EM modeling also permits to analyze the feasibility of brain MWI, which depends on the power transmission from the antennas towards the human brain. In order to estimate this transmission, simplified but still representative models, including intermediate layers -skin, fat, bone, and CerebroSpinal Fluid (CSF) - of the head, are proposed in the framework of simulations (analytical tools) and experimental validations (3D printed head phantom). For the imaging task, the physics of the EM scattering, leads to complex non-linear inverse scattering problems (consisting in retrieving from a set of field measurements the physical parameters which produced them) for which reliable assumptions and approximations must be found. For brain MWI, estimating and quantifying the degree of non-linearity allows for determining the scope of application of existing algorithms, for which different regularizers are applied. Modeling and inverse problem resolution for brain MWI investigated in the present work are ultimately meant to contribute to the development of a technology dedicated to brain stroke detection, differentiation, and monitoring.