  Motivated by periodicity theorems for Real $K$-theory and Grothendieck--Witt
theory and, separately, work of Hori-Walcher on the physics of Landau-Ginzburg
orientifolds, we introduce and study categories of Real matrix factorizations.
Our main results are generalizations of Kn\"{o}rrer periodicity to categories
of Real matrix factorizations. These generalizations are structurally similar
to $(1,1)$-periodicity for $KR$-theory and $4$-periodicity for
Grothendieck-Witt theory. We use techniques from Real categorical
representation theory which allow us to incorporate into our main results
equivariance for a finite group and discrete torsion twists.
