  The nature of unconventional superconductivity is intimately linked to the
microscopic nature of the pairing interactions. In this work, motivated by
cubic heavy fermion compounds with embedded multipolar moments, we
theoretically investigate superconducting instabilities instigated by
multipolar Kondo interactions. Employing multipolar fluctuations (mediated by
RKKY interaction) coupled to conduction electrons via two-channel Kondo and
novel multipolar Kondo interactions, we uncover a variety of superconducting
states characterized by higher-angular momentum Cooper pairs, $J=0,1,2,3$. We
demonstrate that both odd and even parity pairing functions are possible,
regardless of the total angular momentum of the Cooper pairs, which can be
traced back to the atypical nature of the multipolar Kondo interaction that
intertwines conduction electron spin and orbital degrees of freedom. We
determine that different (point-group) irrep classified pairing functions may
coexist with each other, with some of them characterized by gapped and point
node structures in their corresponding quasiparticle spectra. This work lays
the foundation for discovery and classification of superconducting states in
rare-earth metallic compounds with multipolar local moments.
