  We give an exposure to diagrammatic techniques in waveguide QED systems. A
particular emphasis is placed on the systems with delayed coherent quantum
feedback. Specifically, we show that the $N$-photon scattering matrices in
single-qubit waveguide QED systems, within the rotating wave approximation,
admit for a parametrization in terms of $N-1$-photon effective vertex functions
and provide a detailed derivation of a closed hierarchy of generalized
Bethe-Salpeter equations satisfied by these vertex functions. The advantage of
this method is that the above mentioned integral equations hold independently
of the number of radiation channels, their bandwidth, the dispersion of the
modes they are supporting, and the structure of the radiation-qubit coupling
interaction, thus enabling one to study multi-photon scattering problems beyond
the Born-Markov approximation. Further, we generalize the diagrammatic
techniques to the systems containing more than a single emitter by presenting
an exact set of equations governing the generic two and three-photon scattering
operators. The above described theoretical machinery is then showcased on the
example of a three-photon scattering on a giant acoustic atom, recently studied
experimentally [Nat. Phys. 15, 1123 (2019)].
