We investigate two-magnon Raman scattering from the S 1/2 Heisenberg antiferromagnet on the triangular lattice (THAF), considering both isotropic and anisotropic exchange interactions. We find that the Raman intensity for the isotropic THAF is insensitive to the scattering geometry, while both the line profile and the intensity of the Raman response for the anisotropic THAF shows a strong dependence on the scattering geometry. For the isotropic case we present an analytical and numerical study of the Raman intensity including both the efiect of renormalization of the one-magnon spectrum by 1S corrections and final-state magnonmagnon interactions. The bare Raman intensity displays two peaks related to one-magnon van-Hove singularities. We find that 1S self-energy corrections to the one-magnon spectrum strongly modify this intensity profile. The central Raman-peak is significantly enhanced due to plateaus in the magnon dispersion, the high frequency peak is suppressed due to magnon damping, and the overall spectral support narrows considerably. Additionally we investigate final-state interactions by solving the Bethe-Salpeter equation to O(1S). In contrast to collinear antiferromagnets, the non-collinear nature of the magnetic ground state leads to an irreducible magnon scattering which is retarded and non-separable already to lowest order. We show that final-state interactions lead to a rather broad Raman-continuum centered around approximately twice the roton-energy.