Abstract A Hilbert complex is just a complex 0 → D 0 → D 0 D 1 → D 1 ⋯ → D N − 1 D N → 0 , where the D j are closed operators between Hilbert spaces with domain D j and D j + 1 ∘ D j = 0. Although this is a fairly simple object, it reflects surprisingly much of the structure known from elliptic complexes on noncompact manifolds, the main application we have in mind. In this paper we undertake a systematic study of Hilbert complexes and their relationship with elliptic complexes. It turns out that this perspective gives a common structure to various known theorems along with generalizations and extensions. We apply the abstract machinery to the de Rham complex in several singular situations.