Axion insulators are magnetic [[Topological Insulator|topological insulators]] characterized by a non-trivial $\mathbb{Z}_2$ index. This index is protected by [[Inversion Operator|inversion symmetry]]. This is in contrast to conventional [[Topological Insulator|topological insulators]] (i.e., [[Time-Reversal Symmetric Topological Insulator|TRS-TIs]]), where the index is protected by [[Time-Reversal Operator|time-reversal symmetry]]. The $\mathbb{Z}_2$ index arises from the quantization of the axion $\theta$ term to $0$ (trivial) or $\pi$ (topological). More generally, the $\mathbb{Z}_2$ index arises in a magnetic system whose [[Magnetic Point Group|magnetic point group]] (MPG) has either of the following classes of operations: 1. A composition of [[Time-Reversal Operator|time-reversal]] with a proper [[Rotation|rotation]] 2. A composition of [[Inversion Operator|inversion]] with an improper rotation. Class 1 includes the [[Time-Reversal Operator|time-reversal operator]] itself, while class 2 includes the [[Inversion Operator|inversion operator]] itself as well as things like mirror operators. Any magnetic insulator that fits these criteria has an axion $\mathbb{Z}_2$ classification, and if $\theta = \pi$, then the system is a generalized axion insulator.