After a cursory introduction of the basic ideas behind Born's Reciprocal Relativity theory, the geometry of the cotangent bundle of spacetime is studied via the introduction of nonlinear connections associated with certain $nonholonomic$ modifications of Riemann--Cartan gravity within the context of Finsler geometry. A novel gauge theory of gravity in the $8D$ cotangent bundle $ T^*M$ of spacetime is explicitly constructed and based on the gauge group $ SO (6, 2) \times_s R^8$ which acts on the tangent space to the cotangent bundle $ T_{ ( {\bf x}, {\bf p}) } T^*M $ at each point $ ({\bf x}, {\bf p})$. Several gravitational actions involving curvature and torsion tensors and associated with the geometry of curved phase spaces are presented. We conclude with a brief discussion of the field equations, the geometrization of matter, QFT in accelerated frames, {\bf T}-duality, double field theory, and generalized geometry.