  Problems involving quantum impurities, in which one or a few particles are
interacting with a macroscopic environment, represent a pervasive paradigm,
spanning across atomic, molecular, and condensed-matter physics. In this paper
we introduce new variational approaches to quantum impurities and apply them to
the Fr\"{o}hlich polaron -- a quasiparticle formed out of an electron (or other
point-like impurity) in a polar medium, and to the angulon -- a quasiparticle
formed out of a rotating molecule in a bosonic bath. We benchmark these
approaches against established theories, evaluating their accuracy as a
function of the impurity-bath coupling.
