Will present metric theories of gravity with torsion confined to an extra dimension, 1) when the extra dimension is continuous as in the conventional Kaluza-Klein type theories, 2) when the 4-D space-time manifold is extended to a discrete internal space of two discrete points. With a set of minimal constraints, the non-vanishing connection and torsion coefficients are determined in terms of the metric, leading to modified Einstein equations and important physical consequences. In the case of the continuous extra dimension, for instance, homogeneous, FRW isotropic cosmology has possibilities other than the standard big bang cosmology. Also, one finds among spherically symmetric vacuum solutions, positive mass solutions with naked singularity with no event horizon. In the case of the discrete internal space, theory built within the framework of a non-commutatative geometry, one is led to a bi-metric theory of gravity, with a rich variety of physical models, generating a finite mass spectrum and cosmological constant. Extended to include chiral fermions of the standard model, it leads to the possibility of the much sought CP-violating interactions induced by gravity in the early universe.