One of the drawbacks of conventional grand unification scenarios has been
that the unification scale is too high to permit direct exploration. In this
paper, we show that the unification scale can be significantly lowered (perhaps
even to the TeV scale) through the appearance of extra spacetime dimensions.
Such extra dimensions are a natural consequence of string theories with
large-radius compactifications. We show that extra spacetime dimensions
naturally lead to gauge coupling unification at intermediate mass scales, and
moreover may provide a natural mechanism for explaining the fermion mass
hierarchy by permitting the fermion masses to evolve with a power-law
dependence on the mass scale. We also show that proton-decay constraints may be
satisfied in our scenario due to the higher-dimensional cancellation of
proton-decay amplitudes to all orders in perturbation theory. Finally, we
extend these results by considering theories without supersymmetry;
experimental collider signatures; and embeddings into string theory. The latter
also enables us to develop several novel methods of explaining the fermion mass
hierarchy via D-branes. Our results therefore suggest a new approach towards
understanding the physics of grand unification as well as the phenomenology of
large-radius string compactifications.Comment: 65 pages, LaTeX, 20 figure