We present a novel theory of polaritons which contains the crucial tools to securely tackle the many-body physics of polaritons. Microcavitly polaritons are central to exciting Bose-Einstein condensation and parametric scattering experiments in semiconductors. Besides Coulomb and Pauli scattering (dressed by the relevant exciton-Hopfield coefficients), it beautifully unravels a novel fundamental scattering between two polaritons which can be physically associated with a photon-assisted exchange process, without any Coulomb contribution. The non-trivial consequences of the polariton composite nature - here treated exactly - lead to results noticeably different from the ones of the conventional approaches in which polaritons are mapped into elementary bosons. As an example we compute the scattering rate of two pump polaritons in a typical stimulated scattering experiment. To evaluate the limits of bosonization techniques we also compare qualitatively and quantitatively the scattering rate obtained through the main bosonized approach used up to now in the field to the exact result, and show in particular that unphysical spurious terms appear with bosonization.