Simply put, a Stein manifold is a closed complex submanifold of complex Euclidean space. Stein manifolds have rich function theory and many of their global properties can be understood with the help of sheaf cohomology. As a long term goal we would like to prove vanishing theorems for closed submanifolds of Hilbert spaces, and give a cohomological characterization of these submanifolds. In this talk we survey the available results in this direction and look at the possible near future.