  To every projective variety $X$, we associate a family of hypersurfaces in
different Grassmannians, called the coisotropic hypersurfaces of $X$. These
include the Chow form and the Hurwitz form of $X$. Gel'fand, Kapranov and
Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on
tangent spaces. We present a new and simplified proof of that result. We show
that the coisotropic hypersurfaces of $X$ equal those of its projectively dual
variety, and that their degrees are the polar degrees of $X$. Coisotropic
hypersurfaces of Segre varieties are defined by hyperdeterminants, and all
hyperdeterminants arise in that manner. We derive new equations for the Cayley
variety which parametrizes all coisotropic hypersurfaces of given degree in a
fixed Grassmannian. We provide a Macaulay2 package for transitioning between
$X$ and its coisotropic hypersurfaces.
