  We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings.
In the characteristic $p$-theory we describe the kernel and image of the
$q$-expansion map and prove the existence of filtration for Hilbert modular
forms; we define operators $U$, $V$ and $\Theta_\chi$ and study the variation
of the filtration under these operators. In particular, we prove that every
ordinary eigenform has filtration in a prescribed box of weights. Our methods
are geometric -- comparing holomorphic Hilbert modular forms with rational
functions on a moduli scheme with level-$p$ structure, whose poles are
supported on the non-ordinary locus.
  In the $p$-adic theory we study congruences between Hilbert modular forms.
This applies to the study of congruences between special values of zeta
functions of totally real fields. It also allows us to define $p$-adic Hilbert
modular forms "\`a la Serre" as $p$-adic uniform limit of classical modular
forms, and compare them with the $p$-adic modular forms "\`a la Katz" that are
regular functions on a certain formal moduli scheme. We show that the two
notions agree for cusp forms and for a suitable class of weights containing all
the classical ones. We extend the operators $V$ and $\Theta_\chi$ to the
$p$-adic setting.
