Many invertible actions $tau$ on a set ${mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${mathcal{S}}$, exhibit the following property which we dub textbf{homomesy}: the average of $f$ over each $tau$-orbit in ${mathcal{S}}$ is the same as the average of $f$ over the whole set ${mathcal{S}}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushevs conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suters action on certain subposets of Youngs Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.
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