  When a mobile impurity interacts with a surrounding bath of bosons, it forms
a polaron. Numerous methods have been developed to calculate how the energy and
the effective mass of the polaron are renormalized by the medium for
equilibrium situations. Here we address the much less studied non-equilibrium
regime and investigate how polarons form dynamically in time. To this end, we
develop a time-dependent renormalization group approach which allows
calculations of all dynamical properties of the system and takes into account
the effects of quantum fluctuations in the polaron cloud. We apply this method
to calculate trajectories of polarons following a sudden quench of the
impurity-boson interaction strength, revealing how the polaronic cloud around
the impurity forms in time. Such trajectories provide additional information
about the polaron's properties which are challenging to extract directly from
the spectral function measured experimentally using ultracold atoms. At strong
couplings, our calculations predict the appearance of trajectories where the
impurity wavers back at intermediate times as a result of quantum fluctuations.
Our method is applicable to a broader class of non-equilibrium problems. We
also apply it to calculate the spectral function and find good agreement with
experimental results. At very strong couplings, we predict that quantum
fluctuations lead to the appearance of a dark continuum with strongly
suppressed spectral weight at low energies. While our calculations start from
an effective Fr\"ohlich Hamiltonian describing impurities in a
three-dimensional Bose-Einstein condensate, we also calculate the effects of
additional terms in the Hamiltonian beyond the Fr\"ohlich paradigm. We
demonstrate that the main effect of these additional terms on the attractive
side of a Feshbach resonance is to renormalize the coupling strength of the
effective Fr\"ohlich model.
