  This paper contains an attempt to formulate rigorously and to check
predictions in enumerative geometry of curves following from Mirror Symmetry.
The main tool is a new notion of stable map. We give an outline of a
contsruction of Gromov-Witten invariants for all algebraic projective or closed
symplectic manifolds. Mirror Symmetry in the basic example of rational curves
on a quintic 3-folds is reduced to certain complicated but explicit identity.
The strategy of computations can be described as follows: 1) we reduce counting
problems to questions concerning Chern classes on spaces of curves on the
ambient projective space, 2) using Bott's residue formula we pass to the space
of (degenerate) curves invariant under the action of the group of diagonal
matrices, 3) we get a sum over trees and evaluate it using the technique of
Feynman diagrams. Our computation scheme gives ``closed'' formulas for
generating functions in topological sigma-model for a wide class of manifolds,
covering many Calabi-Yau and Fano varieties.

High Energy Physics - Theory;Algebraic Geometry;Algebraic Geometry