  This Thesis reviews an extension of the MSSM by the addition of an anomalous
abelian vector multiplet and contains some original results concerning the
phenomenology of an anomalous $Z'$. The review part covers an introduction of
the MSSM focusing on its main features, a discussion on the chiral anomalies
and how to cancel them in the Standard Model and by the Green-Schwarz
mechanism. Then, the original results are presented. We build the Lagrangian
for the Minimal Anomalous $U(1)'$ Extension of the MSSM where the anomalies are
cancelled by the Green-Schwarz mechanism and the addition of Chern-Simons
terms, stressing the main differences between our model and the MSSM. The
advantage of this choice over the standard one is that it allows for arbitrary
values of the quantum numbers of the extra U(1). As a first step towards the
study of hadron annihilations producing four leptons in the final state (a
clean signal which might be studied at LHC) we then compute the decays $Z'\to
Z_0 \g$ and $Z'\to Z_0 Z_0$. We find that the largest values of the decay rate
are $\sim 10^{-4}$ GeV, while the expected number of events per year at LHC is
at most of the order of 10. Then we compute the relic density predicted by our
model with a new dark matter candidate, the axino, which is the LSP of the
theory. We find agreement with experimental data admitting a bino-higgsino NLSP
or a wino-like NLSP, almost degenerate in mass to the axino.
