  Symmetry-protected topological (SPT) phases are gapped short-range-entangled
quantum phases with a symmetry $G$, which can all be smoothly connected to the
trivial product states if we break the symmetry. It has been shown that a large
class of interacting bosonic SPT phases can be systematically described by
group cohomology theory. In this paper, we introduce a (special) group
supercohomology theory which is a generalization of the standard group
cohomology theory. We show that a large class of short-range interacting
fermionic SPT phases can be described by the group supercohomology theory.
Using the data of super cocycles, we can obtain the ideal ground state wave
function for the corresponding fermionic SPT phase. We can also obtain the bulk
Hamiltonian that realizes the SPT phase, as well as the anomalous (ie,
non-on-site) symmetry for the boundary effective Hamiltonian. The anomalous
symmetry on the boundary implies that the symmetric} boundary must be gapless
for 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. As
an application of this general result, we construct a new SPT phase in 3D, for
interacting fermionic superconductors with coplanar spin order (which have
$T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetries
described by a full symmetry group $Z_2^T\times Z_2^f$). Such a fermionic SPT
state can neither be realized by free fermions nor by interacting bosons
(formed by fermion-pairs), and thus are not included in the K-theory
classification for free fermions or group cohomology description for
interacting bosons. We also construct three interacting fermionic SPT phases in
2D with a full symmetry group $Z_2\times Z_2^f$. Those 2D fermionic SPT phases
all have central-charge $c=1$ gapless edge excitations, if the symmetry is not
broken.
