Understanding the physics of a mobile impurity in a quantum system is a long
standing problem. Indeed the quantum bath is strongly affecting the properties of the
impurity leading at best to renormalized parameters, such as the mass, and
potentially to radically new physics. I will discuss this problem for the case of a one dimensional
bath for which the Luttinger nature of the bath leads to a novel set of properties, such as
sub-diffusion for the impurity. I will also discuss such physics in the light of the
recent experiments on cold atomic gases targeted to probe these issues, and in particular the
ones of the Florence and Munich group.