The Resolution of the Identity (RI) is widely used in many-body algorithms. It expresses the completeness of a set of functions that possess the familiar orthonormality property. In the first part of my lecture, I will discuss functions that possess an analogous property called Coulomb-orthonormality and which permit us to resolve the two-particle Coulomb operator into a sum of products of one-particle functions. Connections to Cholesky decomposition and Kronecker-product approximation will be made. In the second part of the lecture, I will present and discuss numerical applications of Coulomb resolution in the context of electronic structure theory.