  We use a non-perturbative renormalization group approach to develop a unified
picture of the Bose polaron problem, where a mobile impurity is strongly
interacting with a surrounding Bose-Einstein condensate (BEC). A detailed
theoretical analysis of the phase diagram is presented and the
polaron-to-molecule transition is discussed. For attractive polarons we argue
that a description in terms of an effective Fr\"ohlich Hamiltonian with
renormalized parameters is possible. Its strong coupling regime is realized
close to a Feshbach resonance, where we predict a sharp increase of the
effective mass. Already for weaker interactions, before the polaron mass
diverges, we predict a transition to a regime where states exist below the
polaron energy and the attractive polaron is no longer the ground state. On the
repulsive side of the Feshbach resonance we recover the repulsive polaron,
which has a finite lifetime because it can decay into low-lying molecular
states. We show for the entire range of couplings that the polaron energy has
logarithmic corrections in comparison with predictions by the mean-field
approach. We demonstrate that they are a consequence of the polaronic mass
renormalization which is due to quantum fluctuations of correlated phonons in
the polaron cloud.
