Symplectic manifolds are at once flexible and rigid--they admit many automorphisms, but with striking restrictions when compared to the space of self diffeomorphisms. The goal of this talk is to demonstrate the flexibility of symplectic manifolds when compared to more classical geometries (e.g. Riemannian geometry), and the relative rigidity symplectic manifolds exhibit when compared to the category of smooth manifolds. Symplectic topology arises as the study of the interplay between this flexibility and rigidity.