We consider the motion of an impurity particle in a general one-dimensional quantum fluid at zero temperature. The dispersion relation \( \Omega(P) \) of the impurity is strongly affected by interactions with the fluid as the momentum approaches \(\pm \pi \hbar n\), \(\pm 3\pi \hbar n\), …, where \( n\) is the density. This behavior is caused by singular \(\pm 2\pi \hbar n\) scattering processes and can be understood by analogy to the Kondo effect, both at strong and weak couplings, with the possibility of a quantum phase transition where \(\Omega(\pm \pi \hbar n)\) jumps to zero with increasing coupling. The low energy singularities in the impurity spectral function can be understood on the same footing.