Given a word $w \in F_{r}$ in a free group on $r$-generators, and
a finite group $G$, the word map $G^{r} \to G$ is the map obtained
by evaluating the word $w$ on the $r$-tuple of elements of $G$.
The word map yields a measure on the finite group $G$ by pushing
forward the uniform measure on $G^{r}$. We will discuss these
measures and their connection to conjectures regarding profinite rigidity
of words in free groups. We will then describe some techniques used
to prove special cases of these conjectures by studying word measures
on symmetric groups. If time permits, we will also discuss how these
ideas are related to solutions of equations in free groups and the
non-negative immersions of graphs of free groups with cyclic edge
groups.