  One of the most challenging problems in solid state systems is the
microscopic analysis of electronic correlations. A paramount minimal model that
encodes correlation effects is the Hubbard Hamiltonian, which -- albeit its
simplicity -- is exactly solvable only in a few limiting cases and approximate
many-body methods are required for its solution. In this review we present an
overview on the non-perturbative Two-Particle Self-Consistent method (TPSC)
which was originally introduced to describe the electronic properties of the
single-band Hubbard model. We introduce here a detailed derivation of the
multi-orbital generalization of TPSC and discuss particular features of the
method on exemplary interacting models in comparison to dynamical mean-field
theory results.
