  In this paper we study the relationship between two different
compactifications of the space of vector bundle quotients of an arbitrary
vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is
a moduli space of stable maps to the relative Grassmannian.
  We establish an essentially optimal upper bound on the dimension of the two
compactifications. Based on that, we prove that for an arbitrary vector bundle,
the Quot schemes of quotients of large degree are irreducible and generically
smooth. We precisely describe all the vector bundles for which the same thing
holds in the case of the moduli spaces of stable maps. We show that there are
in general no natural morphisms between the two compactifications. Finally, as
an application, we obtain new cases of a conjecture on effective base point
freeness for pluritheta linear series on moduli spaces of vector bundles.
