  The prime goal of this thesis is to study anomalous gauge boson self
couplings, triple gauge boson couplings in particular, with the help of spin
polarization observables of the gauge bosons $Z$ and $W^\pm$ in the presence of
beam polarizations where ever possible. The neutral triple gauge boson
couplings, i.e., $ZZZ$, $ZZ\gamma$, $Z\gamma\gamma$, are studied in
$ZZ/Z\gamma$ ($2lq\bar{q}/2l\gamma$) production at an $e^+e^-$ collider with
and without beam polarization. Some of these anomalous couplings are also
studied in $ZZ$ ($4l$) production at the LHC. In the charge sector the
anomalous gauge boson couplings, i.e., $WWZ$, $WW\gamma$ have been studied at
an $e^+e^-$ collider in $W^+W^-$ ($l^-\bar{\nu_l}q^\prime\bar{q}$) production.
The $WWZ$ anomalous couplings are also studied in $ZW^\pm$ production at LHC in
$3l$ + missing $E_T$ channel. All the analyses at an $e^+$-$e^-$ collider have
been performed for center-of-mass (CM) energy of $500$ GeV and integrated
luminosity of $100$ fb$^{-1}$. The analyses at the LHC are performed at $13$
TeV CM energy of $pp$ collisions. The cross sections and polarization
asymmetries, along with other asymmetries (forward-backward, azimuthal), are
used to obtain simultaneous limits on the anomalous couplings using
Markov-Chain--Monte-Carlo (MCMC) method in each process. The polarization
asymmetries can distinguish between $CP$-even and $CP$-odd couplings and help
to put tighter constraints on the couplings. The polarization of the initial
$e^-$ and $e^+$ beam, in case of $e^+e^-$ collider, are used to increase the
signal to background ratio, putting tighter constraints on the anomalous
couplings. The polarization asymmetries are instrumental in the measurement of
anomalous couplings should a deviation from the SM be observed.
