  While calculations and measurements of single-particle spectral properties
often offer the most direct route to study correlated electron systems, the
underlying physics may remain quite elusive, if information at higher particle
levels is not explicitly included. Here, we present a comprehensive overview of
the different approaches which have been recently developed and applied to
identify the dominant two-particle scattering processes controlling the shape
of the one-particle spectral functions and, in some cases, of the physical
response of the system. In particular, we will discuss the underlying general
idea, the common threads and the specific peculiarities of all the proposed
approaches. While all of them rely on a selective analysis of the
Schwinger-Dyson (or the Bethe-Salpeter) equation, the methodological
differences originate from the specific two-particle vertex functions to be
computed and decomposed. Finally, we illustrate the potential strength of these
methodologies by means of their applications the two-dimensional Hubbard model,
and we provide an outlook over the future perspective and developments of this
route for understanding the physics of correlated electrons.
