The Kondo effect, wherein electrons in a metal scatter off of a local dipole moment, can have profound effects on the nature of the metallic state, including the generation of non-Fermi liquid states lacking well-defined quasiparticles. In heavy fermion compounds, the Kondo effect competes with magnetic ordering, leading to quantum critical points between Fermi/non-Fermi liquids and magnetic order. Multipolar local moments add an exciting ingredient to this picture, and offer new avenues to exotic Kondo physics, multipolar order, and quantum criticality. In this thesis presentation, I will discuss a very interesting example of a compound containing dipolar, quadrupolar, and octupolar moments, and the quantum critical points between multipolar ordered states and Fermi/non-Fermi liquids. Using renormalization group techniques, I show how two consecutive quantum phase transitions may arise, accompanied by jumps in the Fermi surface size, and the disappearance of multipolar orders. I will also discuss connection to recent experiments.