Consider a large random structure—a stochastic process on the line, a random graph, a random field on the grid—and a function that depends only on a small part of the structure. Now use elements of a transformation group to ‘move’ the domain of the function over the structure, and average over the collected values. It is known from ergodic theory that such averages converge to (conditional) expectations, if (i) the transformations leave the distribution invariant and (ii) the group is sufficiently nice. I will present results that show they are also asymptotically normal, under a suitable mixing condition. Several known central limit theorems for stationary random fields, graphon models, etc emerge as special cases.