The Bose polaron is a system in which few impurities are immersed in a cold gas of bosonic atoms. It is of interest in both mathematics and physics as a model problem where approaches to more general polaron systems in condensed matter can be tested, experimentally or by rigorous analysis. The Bogoliubov-Fröhlich Hamiltonian provides a description of the Bose polaron by impurities interacting with a field of excitation quasi-particles. I will discuss its derivation as an effective Hamiltonian from many-body quantum mechanics in the mean-field (high density, weakly interacting) regime (joint work with Peter Pickl). I will also mention some work in progress on extensions to dilute gases (joint work with Arnaud Triay).