The idea that the physical spacetime might have more than four dimensions dates back to almost a century ago, but during the last few decades it has received renewed attention, due to its connections to String Theory, and to potential applications for New Physics at energies within reach of the LHC experiments. In particular, gauge theories defined in a spacetime with compact extra dimensions are a theoretically appealing setup for models of Grand Unification and electroweak symmetry breaking, for the fermion hierarchy problem, and for the strong-CP problem. While a gauge theory in a spacetime with more than four dimensions is generally non-renormalizable, and thus cannot be considered as a fundamental theory, it can be interpreted as a low-energy effective description of an underlying, more fundamental theory. In this talk, after reviewing the general properties of models defined in extra dimensions, I shall discuss the construction and the non-perturbative lattice investigation of a strongly coupled SU(2) gauge model in a five-dimensional spacetime with a compact extra dimension. In particular, I shall present the phase diagram of the model, and discuss how various four-dimensional theories can emerge in different low-energy limits.