  We state the Brumer-Stark conjecture and motivate it from two perspectives.
Stark's perspective arose in his attempts to generalize the classical Dirichlet
class number formula for the leading term of the Dedekind zeta function at
$s=1$ (equivalently, $s=0$). Brumer's perspective arose by generalizing
Stickelberger's work regarding the factorization of Gauss sums and the
annihilation of class groups of cyclotomic fields. These viewpoints were
synthesized by Tate, who stated the Brumer-Stark conjecture in its current
form.
  The conjecture considers a totally real field $F$ and a finite abelian CM
extension $H/F$. It states the existence of $p$-units in $H$ whose valuations
at places above $p$ are related to the special values of the $L$-functions of
the extension $H/F$ at $s=0$. Essentially equivalently, the conjecture states
that a Stickelberger element associated to $H/F$ annihilates the (appropriately
smoothed) class group of $H$. This conjecture has been refined by many authors
in multiple directions.
  We conclude by stating our results toward these various conjectures and
summarizing the proofs. In particular, we prove the Brumer-Stark conjecture,
Rubin's higher rank version, and Kurihara's conjecture, all "away from 2." We
also prove strong partial results toward Gross's conjecture and the exact
$p$-adic analytic formula for Brumer-Stark units. The key technique involved in
the proofs is Ribet's method. We demonstrate congruences between Hilbert
modular Eisenstein series and cusp forms, and use the associated Galois
representations to construct Galois cohomology classes. These cohomology
classes are interpreted in terms of Ritter-Weiss modules, from which results on
class groups may be deduced.
