  A moduli space of stable quotients of the rank n trivial sheaf on stable
curves is introduced. Over nonsingular curves, the moduli space is
Grothendieck's Quot scheme. Over nodal curves, a relative construction is made
to keep the torsion of the quotient away from the singularities. New
compactifications of classical spaces arise naturally: a nonsingular and
irreducible compactification of the moduli of maps from genus 1 curves to
projective space is obtained. Localization on the moduli of stable quotients
leads to new relations in the tautological ring generalizing Brill-Noether
constructions.
  The moduli space of stable quotients is proven to carry a canonical 2-term
obstruction theory and thus a virtual class. The resulting system of descendent
invariants is proven to equal the Gromov-Witten theory of the Grassmannian in
all genera. Stable quotients can also be used to study Calabi-Yau geometries.
The conifold is calculated to agree with stable maps. Several questions about
the behavior of stable quotients for arbitrary targets are raised.
