  We study fermionic matrix product operator algebras and identify the
associated algebraic data. Using this algebraic data we construct fermionic
tensor network states in two dimensions that have non-trivial
symmetry-protected or intrinsic topological order. The tensor network states
allow us to relate physical properties of the topological phases to the
underlying algebraic data. We illustrate this by calculating defect properties
and modular matrices of supercohomology phases. Our formalism also captures
Majorana defects as we show explicitly for a class of $\mathbb{Z}_2$
symmetry-protected and intrinsic topological phases. The tensor networks states
presented here are well-suited for numerical applications and hence open up new
possibilities for studying interacting fermionic topological phases.
