  We discuss the two-channel Kondo problem with a pseudogap density of states,
$\rho(\w)\propto|\w|^r$, of the bath fermions. Combining both analytical and
numerical renormalization group techniques, we characterize the impurity phases
and quantum phase transitions of the relevant Kondo and Anderson models. The
line of stable points, corresponding to the overscreened non-Fermi liquid
behavior of the metallic $r=0$ case, is replaced by a stable particle-hole
symmetric intermediate-coupling fixed point for $0<r<\rmax\approx0.23$. For
$r>\rmax$, this non-Fermi liquid phase disappears, and instead a critical fixed
point with an emergent spin--channel symmetry appears, controlling the quantum
phase transition between two phases with stable spin and channel moments,
respectively. We propose low-energy field theories to describe the quantum
phase transitions, all being formulated in fermionic variables. We employ
epsilon expansion techniques to calculate critical properties near the critical
dimensions $r=0$ and $r=1$, the latter being potentially relevant for
two-channel Kondo impurities in neutral graphene. We find the analytical
results to be in excellent agreement with those obtained from applying Wilson's
numerical renormalization group technique.
