In infinite measure-preserving dynamical systems, classical limit theorems such as Birkhoff’s theorem or  CLTs no longer apply. In this talk, I will present the main framework of limit laws in infinite ergodic theory. In particular, I will focus on some generalized stable laws that arise in this setting. I will show how these new limits emerge naturally from the structure of the dynamics. They can be obtained through the study of limits for rare-event point processes (particularly spatio-temporal ones) and we will also show corresponding results for them in the infinite-measure context.