  In this thesis, we study the hidden symmetries in the SM and also use
discrete gauge symmetries as model building tools to solve various problems in
the SM as well as the MSSM, such as R-parity, mu-term, stabilizing the axion
solutions, etc. The flavor independent non-supersymmetric SM at the
renormalizable level has a discrete Z_3 gauge symmetry known as baryon parity.
It is anomaly free at the discrete level as a result of the existence of three
generations. The symmetry can effectively act as the Baryon number up to the
\Delta B=3 mod 3 level which is also consistent with the prediction from
non-perturbative processes corrections in the SM, such as electroweak instanton
and sphaleron processes. This symmetry is not consistent with the simple GUTs
since those theories explicitly break it. Thus this baryon parity provides a
strong hint for new physics like GUTs. Quantum mechanically, we estimate the
triple nucleon decay rate which is predicted by the existence of this symmetry.
We find a simple U(1) realization with the presence of right-handed neutrinos,
from which this baryon parity can naturally emerge. Gauged R-parity is studied
in the following chapter. We also study various different approaches to the
mu-term problem via a symmetry classification. One explicit example in terms of
a Z_4 subgroup of the anomalous U(1) symmetry is given. Another solution arises
from a SUSY version of QCD invisible axion. Discrete flavor gauge symmetries
are discussed to explain the observed hierarchy fermion masses. D-term
splitting problem can then be avoided. In the last part, we show how to use
discrete gauge symmetries to stabilize the QCD invisible axion solutions, for
both DFSZ and KSVZ invisible axion models.
