Symplectic capacities are measurements of symplectic size.  They are often defined as the lengths of certain periodic trajectories of dynamical systems, and so they connect symplectic embedding problems with dynamics.  I will explain joint work showing how to recover the volume of many symplectic 4-manifolds from the asymptotics of a family of symplectic capacities, called "ECH" capacities.   I will then explain how this asymptotic formula was used by Asaoka and Irie to prove the following dynamical result: for a C^{\infty} generic diffeomorphism of S^2 preserving an area form, the union of periodic points is dense