  The fibre bundle formalism inherent to the construction of non-abelian
Kaluza-Klein theories is presented and its associated dimensional reduction
process analysed: is performed the dimensional reduction of G-invariant matter
and gauge fields over a multidimensional universe; the harmonic decomposition
of non-symmetric fields over its internal space established and their
dimensional reduction done. The spontaneous compactification process is
presented. It is shown that during the dimensional reduction process two types
of symmetry breaking can occur: a geometric followed by a spontaneous symmetry
breaking. This last is connected to a scalar field resulting from the
dimensional reduction process itself. We determine explicitly the scalar
potential form leading to that symmetry breaking for the case in which the
internal space is symmetric and an analysis for the general case is performed.
The principal problems found in the construction of realistic Kaluza-Klein
models are discussed and some solutions reviewed. The hierarchy problem is
examined and its solution within Kaluza-Klein models with a large volume
internal space considered. As an alternative solution, the Randall-Sundrum
model is presented and its principal properties analysed. In special, the
stability of the hierarchy is studied.
