The SO(4) symmetry has been used in various fields in physics. Here we have considered three fields that the SO(4) symmetry group can be applied. The application of SO(4) symmetry in quantum mechanics was realized in the symmetry of hydrogen atom. Using this symmetry we could explain the accidental degeneracy of the hydrogen atom energy levels without solving the Schrödinger equation. Moreover, we also deduced the hydrogen atom spectrum which is given as E, =-722. We gave our results using the matrix representations of the operators. We constructed the quaternionic representations of the I = 0 and 1 = 1 states. Next we applied the SO(4) symmetry to determine the 'He superfluidity phases. First we showed that the normal 'He liquid phase order parameter is invariant under SO(4) OT x U(1)(=SO(3)s x SO(3)ų x U(1)) symmetry group. Thus by using the broken symmetry concept we found all possible subgroups of this full symmetry group that leave the 'He order parameter invariant and which are corresponding to the 'He superfluid phases. We presented the results with a new approach so that the phase transformations are easily understood. SO(4) symmetry was also applied to study the nucleon antinucleon systems from which we could deduce all possible linear relations between the scattering amplitudes of the nucleon and antinucleon scattering processes. The results are presented in a table which has not been obtained elsewhere.