Abstract: This talk is intended to a general audience. Symplectic geometry is at the crossroad of Riemannian geometry, Complex Analysis and Rational Homotopy theory. We briefly review the basic notions and show the links between these subjects. Then we outline some problems we (A.Banyaga, J.B. Gatsinzi, and F. Massamba) are working on, concerning the path from symplectic structures to Kaehler structures, and the link to the notion of formality in Rational Homotopy theory.