  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 Panyushev's 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, Suter's action on certain subposets of
Young's 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.
