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.
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