Numerical lattice calculations provide a powerful method for systematically estimating finite-volume Euclidean correlators. To connect to physical observables, one must analyze the role of the Euclidean signature and finite volume, as well as other systematic effects. In this seminar, I will review work on interpreting multi-hadron observables as spectral functions, given by inverting the Laplace transform on a particular lattice correlator. I will describe methods for regulating this notoriously ill-posed problem by targeting a smeared spectral function. This “smearing" turns out to be of great physical importance, e.g. for defining the infinite-volume limit and for implementing the required i-epsilon pole prescription. I will further discuss how this can be understood as an extension of the famous work of Maiani and Testa on Euclidean correlators. These methods are expected to be most relevant in quantities with many hadronic intermediate states including long distance contributions to heavy meson mixing and decays.