  We calculate the small quantum orbifold cohomology of arbitrary weighted
projective spaces. We generalize Givental's heuristic argument, which relates
small quantum cohomology to S^1-equivariant Floer cohomology of loop space, to
weighted projective spaces and use this to conjecture an explicit formula for
the small J-function, a generating function for certain genus-zero
Gromov-Witten invariants. We prove this conjecture using a method due to
Bertram. This provides the first non-trivial example of a family of orbifolds
of arbitrary dimension for which the small quantum orbifold cohomology is
known. We also obtain formulas for the small J-functions of weighted projective
complete intersections satisfying a combinatorial condition; this condition
naturally singles out the class of orbifolds with terminal singularities.
