A surface $S$ in a complete $3$-manifold $M$ is area-minimizing mod $2$ if it has least area among all surfaces, orientable or nonorientable in the same homology class. These surfaces present a rich and interesting geometry, even in flat or positively curved $3$-manifolds. For instance, if $M$ is flat, Fischer-Colbrie and Schoen, Do Carmo and Peng, and Pogorelov proved that complete two-sided stable minimal surfaces are flat, but Ross proved that some nonorientable quotient of the classic Schwarz P and D surfaces are estable, and we proved that an area minimizing surface in $\mathbb{R}^2\times S^1$ is either planar or a quotient of the Helicoid. We will review some results about this problem and we will prove that area minimizing surfaces in flat $3$-tori are planar.