  The pseudogap Kondo problem, describing quantum impurities coupled to
fermionic quasiparticles with a pseudogap density of states, rho(omega) ~
|omega|^r, shows a rich zero-temperature phase diagram, with different screened
and free moment phases and associated transitions. We analyze both the
particle-hole symmetric and asymmetric cases using renormalization group
techniques. In the vicinity of r=0, which plays the role of a lower-critical
dimension, an expansion in the Kondo coupling is appropriate. In contrast, r=1
is the upper-critical dimension in the absence of particle-hole symmetry, and
here insight can be gained using an expansion in the hybridization strength of
the Anderson model. As a by-product, we show that the particle-hole symmetric
strong-coupling fixed point for r<1 is described by a resonant level model, and
corresponds to an intermediate-coupling fixed point in the renormalization
group language. Interestingly, the value r=1/2 plays the role of a second
lower-critical dimension in the particle-hole symmetric case, and there we can
make progress by a novel expansion performed around a resonant level model. The
different expansions allow a complete description of all critical fixed points
of the models and can be used to compute a variety of properties near
criticality, describing universal local-moment fluctuations at these impurity
quantum phase transitions.
