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The one-dimensionality of carriers in SWNTs results in strong Coulomb interactions, the non-perturbative binding of an electron-hole pair into an exciton being the first striking signature. Besides the excitonic nature of the optical resonances in SWNTs, several time-resolved experiments have shown the evidence that the strong Coulomb correlations also affect the relaxation dynamics in SWNTs. In particular, photoluminescence and pump-probe measurements using ultrafast excitation pulses have revealed that the population relaxation dynamics is driven by an efficient exciton-exciton annihilation (EEA) process, even at low exciton density. This Auger process is one type of exciton-exciton interaction where one exciton recombines while the second one is promotted to a high-energy state. Since EEA is known to play a key role in the population relaxation of Frenkel excitons in organic materials, especially J-aggregates and conjugated polymers, the high efficiency of EEA in SWNTs suggests that excitons in SWNTs are mostly of Frenkel type. In the following, we will show that the EEA process does not account for the excitonic collision-induced broadening measured in SWNTs. Power-dependent measurements of the excitonic homogeneous linewidth demonstrate the predominance of another type of exciton-exciton interaction, namely elastic exciton-exciton scattering (EES). This latter process redistributes the population within exciton bands and contributes to pure dephasing of the excitonic transition. EES is widely observed for Wannier excitons in inorganic bulk semiconductors or quantum wells but not for Frenkel excitons in organic materials, leading to a refined picture of excitons in SWNTs in between Wannier and Frenkel excitons.