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We briefly review some current theoretical and experimental aspects of theproblem of a single spinless impurity in a 3D polarised atomic Fermi gas atzero temperature where the interactions can be tuned using a wide Feshbachresonance. We show that various few-body states in vacuum composed of theimpurity and background gas atoms (single impurity, dimer, trimer, tetramer)give rise to corresponding dressed states ({\em polaron}, {\em dimeron}, {\emtrimeron}, {\em tetrameron}) in the gas and inherit many of theircharacteristics. We study the ground state focussing on the choice of wavefunction and its properties. We raise a few unsolved problems: whether thepolaron and dimeron are really separate branches, what other few-body statesmight exist, the nature of the groundstate for large numbers of particle-holepairs and why is the polaron ansatz so good. We then turn to the excitedstates, and to the calculation of the effective mass. We examine the bounds onthe effective mass and raise a conjecture about that of composite quasiparticlestates.

A single impurity in an ideal atomic Fermi gas: current understanding and some open problems