Resonant Raman scattering has been used for decades to study single walled carbon nanotubes (CNTs), but lacks a consistent theory that simultaneously explains all characteristic signatures. We argue that a proper description requires introducing exciton polaritons as couples excitonic and photonic states. We describe the polaritons by waveguided theory for a nanometre thick cylinder with modified dielectric function. During their propagation along the tube, the polaritons scatter with phonons and are re-emitted as photons with smaller energy (Stokes scattering event). This approach consistently explains the energetic positions of the Raman resonances for the radial-breathing mode (RBM), the G and the 2D line as well as the asymmetry in the scattering intensity of the incoming and outgoing resonances without resorting to higher-order scattering events. We measured the Raman effect on chirality-sorted nanotubes over the first and second exciton resonance and excellently predict its behaviour, which has not been achieved before. The formation of polaritonic states will affect other optical processes in CNTs. Furthermore, exciton-polariton effects appear inevitable in all one-dimensional excitonic systems, similar to our observations in single walled carbon nanotubes.