  Fractional statistics is one of the most intriguing features of topological
phases in 2D. In particular, the so-called non-Abelian statistics plays a
crucial role towards realizing universal topological quantum computation.
Recently, the study of topological phases has been extended to 3D and it has
been proposed that loop-like extensive objects can also carry fractional
statistics. In this work, we systematically study the so-called three-loop
braiding statistics for loop-like excitations for 3D fermionic topological
phases. Most surprisingly, we discovered new types of non-Abelian three-loop
braiding statistics that can only be realized in fermionic systems (or
equivalently bosonic systems with fermionic particles). The simplest example of
such non-Abelian braiding statistics can be realized in interacting fermionic
systems with a gauge group $\mathbb{Z}_2 \times \mathbb{Z}_8$ or $\mathbb{Z}_4
\times \mathbb{Z}_4$, and the physical origin of non-Abelian statistics can be
viewed as attaching an open Majorana chain onto a pair of linked loops, which
will naturally reduce to the well known Ising non-Abelian statistics via the
standard dimension reduction scheme. Moreover, due to the correspondence
between gauge theories with fermionic particles and classifying fermionic
symmetry-protected topological (FSPT) phases with unitary symmetries, our study
also give rise to an alternative way to classify FSPT phases with unitary
symmetries. We further compare the classification results for FSPT phases with
arbitrary Abelian total symmetry $G^f$ and find systematical agreement with
previous studies using other methods. We believe that the proposed framework of
understanding three-loop braiding statistics (including both Abelian and
non-Abelian cases) in interacting fermion systems applies for generic fermonic
topological phases in 3D.
