From the "last geometric theorem" of Poincare to Arnold's conjectures and Gromov's holomorphic curves.
We will describe the development of ideas and methods of symplectic topology from Poincaré to Gromov and our time. Henri Poincare envisioned symplectic topology as an approach of deriving qualitative properties of Hamiltonian systems in Mechanics without explicitly solving the equations of motion. The modern era of the subject, which was stimulated in 1960-70s by V.I. Arnold's conjectures, began in 1980s with a proof of symplectic rigidity and the introduction by M. Gromov of the method of holomorphic curves in symplectic manifolds. Many remarkable discoveries quickly followed.