The $C_2$-equivariant stable stems were first studied by Bredon and Landweber, in the 1960s. From the start, it was clear that these groups exhibited certain periodic behavior closely related to James periodicity for stunted projective spaces. This was made more explicit and extensively applied to computations by Araki and Iriye in the late 1970s / early 1980s. In the past decade, this phenomena has been lifted to $\mathbb{R}$-motivic homotopy theory under the guise of "$\tau$-periodicity", and plays a central role in Behrens and Shah's work relating $\mathbb{R}$-motivic and $C_2$-equivariant homotopy theory.

In this talk, I will review some of the above story, and then explain how similar periodic phenomena occurs in $G$-equivariant stable homotopy theory for an arbitrary finite group $G$.