Elongated atom traps can confine ultracold gases in the quasi-one-dimensional regime. We review both the theory of these atom waveguides and their experimental realizations, with emphasis on the collisions of waveguide-bound particles. Under certain conditions, quasi-one-dimensional gases are well described by integrable one-dimensional many-body models with exact quantum solutions. We review the thermodynamic and correlation properties of one such model that has been experimentally realized, that of Lieb and Liniger. We describe the necessary criteria for integrability and its observable effects. We also consider ways to lift integrability, along with some observable effects of non-integrability.