  In this article we study properly discontinuous actions on Hilbert manifolds
giving new examples of complete Hilbert manifolds with nonnegative,
respectively nonpositive, sectional curvature with infinite fundamental group.
We also get examples of complete infinite dimensional K\"ahler manifolds with
positive holomorphic sectional curvature and infinite fundamental group in
contrast with the finite dimensional case and we classify abelian groups acting
linearly, isometrically and properly discontinuously on Stiefel manifolds.
Finally, we classify homogeneous Hilbert manifolds with constant sectional
curvature.
