In the last two decades we have assisted to a rush towards finding a He3-replacing technology capable of detecting neutrons emitted from fissile isotopes. The demand stems from applications like nuclear war-head screening or preventing illicit traffic of radiological materials. Semiconductor detectors stand among the stronger contenders, particularly those based on materials possessing a wide band gap like silicon carbide. We review the workings of SiC-based neutron detectors, along with several issues related to material properties, device fabrication and testing. The paper summarizes the experimental and theoretical work carried out within the E-SiCure project, co-funded by the NATO SPS Programme. Among the achievements, we have the development of successful Schottky barrier based detectors and the identification of the main carrier life-time-limiting defects in the SiC active areas, either already present in pristine devices or introduced upon exposure to radiation fields. The physical processes involved in neutron detection are described. Material properties as well as issues related to epitaxial growth and device fabrication are addressed. The presence of defects in as-grown material, as well as those introduced by ionizing radiation are reported. We finally describe several experiments carried out at the Jozef Stefan Institute TRIGA Mark II reactor (Ljubljana, Slovenia), where a set of SiC-based neutron detectors were tested, some of which being equipped with a thermal neutron converter layer. We show that despite the existence of large room for improvement, Schottky barrier diodes based on state-of-the-art 4H-SiC are closing the gap regarding the sensitivity offered by gas-based and that of semiconductor detectors.