How can one simplify a general group action by replacing it with a regular group action, that is, one whose orbits all have the same dimension? In this lecture, I will discuss this problem for proper Lie group actions, as well as its generalization to proper Lie groupoids. I will review some classical results and present recent unpublished joint work with Marius Crainic (Utrecht) and David Martínez-Torres (Madrid). In particular, we introduce the notion of a polaroid, a certain class of proper Lie groupoids for which this problem admits an especially elegant solution.