  The spaces of Riemannian metrics on a closed manifold $M$ are studied. On the
space ${\mathcal M}$ of all Riemannian metrics on $M$ the various weak
Riemannian structures are defined and the corresponding connections are
studied. The space ${\mathcal AM}$ of associated metrics on a symplectic
manifold $M,\omega$ is considered in more detail. A natural parametrization of
the space ${\mathcal AM}$ is defined. It is shown, that ${\mathcal AM}$ is a
complex manifold. A curvature of the space ${\mathcal AM}$ and quotient space
${\mathcal AM}/{\mathcal D}_{\omega}$ is found. The finite dimensionality of
the space of associated metrics of a constant scalar curvature with Hermitian
Ricci tensor is shown.
