We consider a class of nonlinear inverse problems, encompassing e.g. Polarimetric Neutron Tomography (PNT), where one seeks to recover a magnetic field by probing it with Neutron beams and measuring the resulting spin change. In recent years there has been great progress on fundamental theoretical questions regarding injectivity and stability properties for PNT and we survey some of the latest results, including a novel range characterisation for the forward map. One of the drivers behind these results is the desire to give rigorous guarantees for the statistical performance of Bayesian algorithms. The talk is based on joint work with Gabriel Paternain and Richard Nickl.