"Let $(X,\mu)$ be a standard probability space and $G\curvearrowright (X,\mu)$ be a measure-preserving action of a group $G$ on $X$.

The general problem that we consider is to understand the structure of measurable tilings $F\odot A = X$ of $X$ by a measurable tile $A \subseteq X$ shifted by a finite set $F \subseteq G$, thus the shifts $f\cdot A$, $f \in F$ partition $X$ up to null sets.

The motivation comes from the theory of (paradoxical) equidecompositions and tilings in $\mathbb{R}^n$.

After a summary of recent results that concern the spheres ${\bf S}^{d-1}$, where the action is given by rotations, and tori ${\bf T}^d$, where the action is given by translations, I will focus on the intersection of these cases, that is, the case of the circle ${\bf S}^1={\bf T}^1$.

Using the structure theorem of Greenfeld and Tao for tilings of $\mathbb{Z}^d$, we show that measurable tilings of the circle can be reduced to tilings of finite cyclic groups.

This is a joint work with Conley and Pikhurko, and Greenfeld, Rozho\v{n} and Tao."