  We consider the ferromagnetic $q$-state Potts model with zero external field
in a finite volume and assume that the stochastic evolution of this system is
described by a Glauber-type dynamics parametrized by the inverse temperature
$\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$,
in which this multi-spin system has $q$ stable equilibria, corresponding to the
configurations where all spins are equal. Focusing on grid graphs with various
boundary conditions, we study the tunneling phenomena of the $q$-state Potts
model. More specifically, we describe the asymptotic behavior of the first
hitting times between stable equilibria as $\beta \to \infty$ in probability,
in expectation, and in distribution and obtain tight bounds on the mixing time
as side-result. In the special case $q=2$, our results characterize the
tunneling behavior of the Ising model on grid graphs.
