  We extend B. Hassett's theory of weighted stable pointed curves ([Has03]) to
weighted stable maps. The space of stability conditions is described
explicitly, and the wall-crossing phenomenon studied. This can be considered as
a non-linear analog of the theory of stability conditions in abelian and
triangulated categories.
  We introduce virtual fundamental classes and thus obtain weighted
Gromov-Witten invariants. We show that by including gravitational descendants,
one obtains an $\LL$-algebra as introduced in [LM04] as a generalization of a
cohomological field theory.
