  This dissertation reviews the Standard Model formalism as well as the Lepton
Flavour Violating (LFV) decay processes which cause its extension, known as the
physics beyond the SM. Firstly, using the experimental bounds on three body LFV
decays, the corresponding bounds on two body LFV decays are reviewed. The
dynamical suppression of three body LFV decays due to momentum dependent
couplings is also reviewed. Secondly, the role of the LFV decays to explain the
LSND excess is discussed in detail, for which the experimental bounds on three
body LFV decays, i.e. \mu -> 3e are used to constraint the coupling
\tilde{g}_{Z_{\mu e}}, which is needed to calculate the anomalous muon decay
\mu -> e\nu_l\bar{\nu}_l. Then comparing the effective coupling of anomalous
muon decay to r>1.6\times 10^{-3} [9809524], it is proved that LFV is not the
correct hypothesis to explain the LSND excess. Finally, LFV decays at loop
order are studied in Seesaw model of neutrino masses [PRL. 86 2502 (2001)]
where the smallness of the Seesaw neutrino mass may be naturally realized with
m_N (mass of right-handed singlet neutrinos) of order 1 TeV. It is shown that
the Higgs mass of a new scalar doublet with lepton number L=-1 needed in the
model has to be larger than 50 TeV to get the branching ratio of \mu -> 3e to
be consistent with the existing bound on \mu -> 3e. This defeats the original
motivation of the model, namely that there is no physics beyond the TeV energy
scale.
