Given a measured group (G,μ) acting on a nice space X, one might ask how to describe the probability measures ν on X which are invariant with respect to the action of G on “average” with respect to μ. Turns out that such a measure ν usually is uniquely defined, depending on the choice of μ! In my talk I will survey recent progress on understanding such measures with respect to actions of isometry groups of hyperbolic spaces and lamplighter groups. In particular, we will discuss further progress regarding obtaining the formulas for the Hausdorff dimensions of such measures. If time allows, I will mention other research directions I have been pursuing recently. Joint work with Eduardo Silva, Nikolay Bogachev and Giulio Tiozzo.