The need to construct a fermion quantum field theory in black-hole spacetimes is an acute one. The study of gravitational collapse necessitates the need of such. In this dissertation, we construct the theory of free fermions living on the static Schwarzschild black-hole and the rotating Kerr black-hole. The construction capitalises upon the fact that both black-holes are stationary axisymmetric solutions to Einstein's equation. A factorisability ansatz is developed whereby simple quantum modes can be found, for such stationary spacetimes with azimuthal symmetry. These modes are then employed for the purposes of a canonical quantisation of the corresponding fermionic theory. At the same time, we suggest that it may be impossible to extend a quantum field theory continuously across an event horizon. This split of a quantum field theory ensures the thermal character of the Hawking radiation. In our case, we compute and prove that the spectrum of neutrinos emitted from a black-hole via the Hawking process is indeed thermal. We also study fermion scattering amplitudes off the Schwarzschild black-hole.
