  We review recent work on continuous quantum phase transitions in impurity
models, both with fermionic and bosonic baths - these transitions are
interesting realizations of boundary critical phenomena at zero temperature.
The models with fermion bath are generalizations of the standard Kondo model,
with the common feature that Kondo screening of the localized spin can be
suppressed due to competing processes. The models with boson bath are related
to the spin-boson model of dissipative two-level systems, where the interplay
between tunneling and friction results in multiple phases. The competition
inherent to all models can generate unstable fixed points associated with
quantum phase transitions, where the impurity properties undergo qualitative
changes. Interestingly, certain impurity transitions feature both
lower-critical and upper-critical "dimensions" and allow for epsilon-type
expansions. We present results for a number of observables, obtained by both
analytical and numerical renormalization group techniques, and make connections
to experiments.
