Abstract: A lot is known for elliptic operators on compact manifolds, such as the Laplacian. One useful fact among many is the Hodge Theorem: every de Rham cohomology class contains a unique harmonic representative. On non-compact manifolds this is no longer true, but the statement can be adapted to an interesting class of complete non-compact manifolds, called asymptotically cylindrical, which was done by Lockhart and McOwen. In the talk, I will review the Hodge theorem for compact manifolds, and then discuss in which sense it carries over to asymptotically cylindrical manifolds. As a toy example, I will carry out computations for the easiest case of an asymptotically cylindrical manifold, namely R^n, which brings out surprising links to the representation theory of SO(n). Exactly the same calculations apply to asymptotically locally Euclidean manifolds.