Abstract: Moduli spaces of objects associated to a space of X are interesting, the Hilbert scheme is a scheme parametrizing closed subschemes of X with a fixed Hilbert polynomial. In particular the simplest case is the Hilbert scheme of points, which parametrizes the closed subschemes of X of fixed finite length. We will look at examples of Hilbert schemes of points on surfaces, the Hilbert Chow morphism, it’s algebraic description, and discuss some of their relations to quiver varieties.