  If a theory has more than one classically stable vacuum, quantum tunneling
and thermal jumps make the transition between the vacua possible. The
transition happens through a first order phase transition started by nucleation
of a bubble of the new vacuum. The outward pressure of the truer vacuum makes
the bubble expand and consequently eat away more of the old phase. In the
presence of gravity this phenomenon gets more complicated and meanwhile more
interesting. It can potentially have important cosmological consequences. Some
aspects of this decay are studied in this thesis. Solutions with different
symmetry than the generically used O(4) symmetry are studied and their actions
calculated. Vacuum decay in a spatial vector field is studied and novel
features like kinky domain walls are presented. The question of stability of
vacua in a landscape of potentials is studied and the possible instability in
large dimension of fields is shown. Finally a compactification of the
Einstein-Maxwell theory is studied which can be a good lab to understand the
decay rates in compactification models of arbitrary dimensions.
