  It has been shown that the symmetry-protected topological (SPT) phases with
finite Abelian symmetries can be described by Chern-Simons field theory. We
propose a topological response theory to uniquely identify the SPT orders,
which allows us to obtain a systematic scheme to classify bosonic SPT phases
with any finite Abelian symmetry group. We point out that even for finite
Abelian symmetry, there exist bosonic SPT phases beyond the current
Chern-Simons theory framework. We also apply the theory to fermionic SPT phases
with $\mathbb{Z}_m$ symmetry and find the classification of SPT phases depends
on the parity of $m$: for even $m$ there are $2m$ classes, $m$ out of which is
intrinsically fermionic SPT phases and can not be realized in any bosonic
system. Finally we propose a classification scheme of fermionic SPT phases for
any finite, Abelian symmetry.
