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The generalisation of the rigid special geometry of the vector multipletquantum moduli space to the case of supergravity is discussed through thenotion of a dynamical Calabi--Yau threefold. Duality symmetries of thismanifold are connected with the analogous dualities associated with thedynamical Riemann surface of the rigid theory. N=2 rigid gauge theories arereviewed in a framework ready for comparison with the local case. As abyproduct we give in general the full duality group (quantum monodromy) for anarbitrary rigid $SU(r+1)$ gauge theory, extending previous explicitconstructions for the $r=1,2$ cases. In the coupling to gravity, R--symmetryand monodromy groups of the dynamical Riemann surface, whose structure wediscuss in detail, are embedded into the symplectic duality group $\Gamma_D$associated with the moduli space of the dynamical Calabi--Yau threefold.Comment: Latex. Version of previous paper with enlarged and revised appendix  35 pages, plain LaTe

A search for non-perturbative dualities of local N = 2 Yang - Mills theories from Calabi - Yau 3-folds