  We overview the main features of mobile impurities moving in one-dimensional
superfluid backgrounds by modeling it as a mobile Josephson junction, which
leads naturally to the periodic dispersion of the impurity. The dissipation
processes, such as radiative friction and quantum viscosity, are shown to
result from the interaction of the collective phase difference with the
background phonons. We develop a more realistic depleton model of an
impurity-hole bound state that provides a number of exact results interpolating
between the semiclassical weakly-interacting picture and the strongly
interacting Tonks-Girardeau regime. We also discuss the physics of a trapped
impurity, relevant to current experiments with ultra cold atoms.
