  Let $M(\chi)$ denote the maximum of $|\sum_{n\le N}\chi(n)|$ for a given
non-principal Dirichlet character $\chi \pmod q$, and let $N_\chi$ denote a
point at which the maximum is attained. In this article we study the
distribution of $M(\chi)/\sqrt{q}$ as one varies over characters $\pmod q$,
where $q$ is prime, and investigate the location of $N_\chi$. We show that the
distribution of $M(\chi)/\sqrt{q}$ converges weakly to a universal distribution
$\Phi$, uniformly throughout most of the possible range, and get (doubly
exponential decay) estimates for $\Phi$'s tail. Almost all $\chi$ for which
$M(\chi)$ is large are odd characters that are $1$-pretentious. Now,
$M(\chi)\ge |\sum_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q}
|L(1,\chi)|$, and one knows how often the latter expression is large, which has
been how earlier lower bounds on $\Phi$ were mostly proved. We show, though,
that for most $\chi$ with $M(\chi)$ large, $N_\chi$ is bounded away from $q/2$,
and the value of $M(\chi)$ is little bit larger than $\frac{\sqrt{q}}{\pi}
|L(1,\chi)|$.
