In Einstein's gravitational theory, the space-time is pseudo-Riemannian, that is, it has vanishing torsion and vanishing non-metricity. In this work we investigate theories of gravitation in terms of non-Riemannian space-time geometries, then couple electromagnetic and spinor fields to this non-Riemannian gravity. In addition, we treat the Dirac equation containing torsion and non-metricity of space-time by lifting the tangent bundle to the spinor bundle. In order to gain physical insights on torsion and non-metricity we look at the low energy limit of the Dirac equation via the...