  We present the Hamiltonian formalism for the Euler equation of symplectic
fluids, introduce symplectic vorticity, and study related invariants. In
particular, this allows one to extend D.Ebin's long-time existence result for
geodesics on the symplectomorphism group to metrics not necessarily compatible
with the symplectic structure. We also study the dynamics of symplectic point
vortices, describe their symmetry groups and integrability.
