Rings of polynomial invariants of finite group actions are among the most classical objects in commutative algebra. There are many beautiful theorems ensuring that the invariant ring has good properties when the order of the group is invertible. However, if the order of the group is not a unit (i.e., is divisible by the characteristic of the ground field), many of these properties become more subtle. In this talk, I aim to illustrate some of the differences in invariant theory in this setting, and to describe some of my work in progress in this area.