  We develop a microscopic theory of a quantum impurity propagating in a
one-dimensional Bose liquid. As a result of scattering off thermally excited
quasiparticles, the impurity experiences the friction. We find that, at low
temperatures, the resulting force scales either as the fourth or the eighth
power of temperature, depending on the system parameters. For temperatures
higher than the chemical potential of the Bose liquid, the friction force is a
linear function of temperature. Our approach enables us to find the friction
force in the crossover region between the two limiting cases. In the integrable
case, corresponding to the Yang-Gaudin model, the impurity becomes transparent
for quasiparticles and thus the friction force is absent. Our results could be
further generalized to study other kinetic phenomena.
