  This thesis considers fillable contact structures on odd-dimensional
manifolds. For that purpose, Rabinowitz-Floer homology (RFH) is used which was
introduced by Cieliebak and Frauenfelder in 2009. A major part of the thesis is
devoted to technical problems in the definition of RFH. In particular, it is
shown that the moduli spaces involved are cut out transversally. Moreover, it
is proved that RFH is essentially invariant under subcritical handle
attachment. Finally, RFH is calculated for some Brieskorn manifolds. The
obtained results are then used to show for every manifold, which supports
fillable contact structures, that there exist either infinitely many different
fillable contact structures, or one contact structure with infinitely many
different fillings or for every fillable contact structure holds that RFH is
infinite dimensional in every degree.
