In purely algebraic context, Parshall and Wang introduced a notion of normal quantum subgroups for all Hopf algebras using adjoint coactions. In the setting of compact quantum groups, I introduced another notion of normal quantum subgroups using representation theory. I will first show that these two notions of normality are equivalent for compact quantum groups. As applications, I will introduce a quantum analog of the third fundamental isomorphism theorem for groups, which is used along with the equivalence theorem to obtain results on structure of quantum groups with property F and quantum groups with property FD. Other results on normal quantum subgroups for tensor products, free products and crossed products will also be introduced if time permits.