Summary: For a connected symplectic manifold \(X\), we determine the universal central extension of the Lie algebra \(\mathrm{ham}(X)\) of hamiltonian vector fields. We classify the central extensions of \(\mathrm{ham}(X)\), of the Lie algebra \(\mathrm{sp}(X)\) of symplectic vector fields, of the Poisson Lie algebra \(C^\infty (X)\), and of its compactly supported version \(C^\infty_c (X)\).