Gromov-Witten theory is the enumerative theory of curves on
algebraic
varieties. Many important developments in this field, such as mirror
symmetry and integrability results, are based on the Atiyah-Bott
localization theorem. In this talk we will formulate a series of
conjectural results in Gromov-Witten theory obtained via extremal
transitions and large N duality, emphasizing their relation with
the standard localization approach.