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 Foldy-Wouthuysen transformation. Finally we apply all these mathematical concepts to a physical phenomenon; solar neutrino problem. This problem is solved by the assumption of neutrino oscillations and it is seen that the torsion of space-time contributes to the oscillations depending on the polarization of spin states of mass eigenstates and this contribution is of the order of the Planck scale. This book is an extended version of my Ph.D. thesis completed under the supervision of Prof.Dr. Tekin DEREL? in the Middle East Technical University, Ankara, Turkey at 2001.