  We provide a bound on the $\Theta$-regularity of an arbitrary reduced and
irreducible curve embedded in a polarized abelian variety in terms of its
degree and codimension. This is an "abelian" version of
Gruson-Lazarsfeld-Peskine's bound on the Castelnuovo--Mumford regularity of a
non-degenerate curve embedded in a projective space. As an application, we
provide a Castelnuovo type bound for the genus of a curve in a (non necessarily
principally) polarized abelian variety. Finally, we bound the
$\Theta$-regularity of a class of higher dimensional subvarieties in Jacobian
varieties, i.e. the Brill-Noether loci associated to a Petri general curve,
extending earlier work of Pareschi-Popa.
