  The main results in this paper are about the full coalescence time
$\mathsf{C}$ of a system of coalescing random walks over a finite graph $G$.
Letting $\mathsf{m}(G)$ denote the mean meeting time of two such walkers, we
give sufficient conditions under which $\mathbf{E}[\mathsf{C}]\approx
2\mathsf{m}(G)$ and $\mathsf{C}/\mathsf{m}(G)$ has approximately the same law
as in the "mean field" setting of a large complete graph. One of our theorems
is that mean field behavior occurs over all vertex-transitive graphs whose
mixing times are much smaller than $\mathsf{m}(G)$; this nearly solves an open
problem of Aldous and Fill and also generalizes results of Cox for discrete
tori in $d\geq2$ dimensions. Other results apply to nonreversible walks and
also generalize previous theorems of Durrett and Cooper et al. Slight
extensions of these results apply to voter model consensus times, which are
related to coalescing random walks via duality. Our main proof ideas are a
strengthening of the usual approximation of hitting times by exponential random
variables, which give results for nonstationary initial states; and a new
general set of conditions under which we can prove that the hitting time of a
union of sets behaves like a minimum of independent exponentials. In
particular, this will show that the first meeting time among $k$ random walkers
has mean $\approx\mathsf{m}(G)/\bigl({\matrix{k 2}}\bigr)$.
