  When the ground state of a localized ion is a non-Kramers doublet, such
localized ions may carry multipolar moments. For example, Pr$^{3+}$ ions in a
cubic environment would possess quadrupolar and octupolar, but no magnetic
dipole, moments. When such multipolar moments are placed in a metallic host,
unusual interactions between these local moments and conduction electrons
arise, in contrast to the familiar magnetic dipole interactions in the classic
Kondo problem. In this work, we consider the interaction between a single
quadrupolar-octupolar local moment and conduction electrons with $p$-orbital
symmetry as a concrete model for the multipolar Kondo problem. We show that
this model can be written most naturally in the spin-orbital entangled basis of
conduction electrons. Using this basis, the perturbative renormalization group
(RG) fixed points are readily identified. There are two kinds of fixed points,
one for the two-channel Kondo and the other for a novel fixed point. We
investigate the nature of the novel fixed point non-perturbatively using
non-abelian bosonization, current algebra and conformal field theory
approaches. It is shown that the novel fixed point leads to a, previously
unidentified, non-Fermi liquid state with entangled spin and orbital degrees of
freedom, which shows resistivity $\rho \sim T^{\Delta}$ and diverging specific
heat coefficient $C/T \sim T^{-1 + 2\Delta}$ with $\Delta=1/5$. Our results
open up the possibility of myriads of non-Fermi liquid states, depending on the
choices of multipolar moments and conduction electron orbitals, which would be
relevant for many rare-earth metallic systems.
