This paper reviews experimental evidence and the related theoretical background on exotic phenomena arising from local quadrupolar degrees of freedom. Recent extensive studies on praseodymium-based cubic systems, PrT2X20, have revealed that the active quadrupoles in the non-Kramers doublet ground state play a vital role in exhibiting quadrupole orders and superconductivity with underlying peculiar normal paramagnetic electronic states. We focus on four prototype compounds of PrT2X20 (T = Ir, Rh, X = Zn; T = V, Ti, X = Al). Detailed comprehensive comparisons of these compounds have revealed a universal feature of the non-Fermi liquid state emerging from a lattice quadrupolar Kondo effect, and the commonality and individuality of the quadrupolar and superconducting phases. It may be possible to develop a new class of heavy-fermion systems beyond the classic view of heavy fermions on the basis of a concrete understanding of these phenomena.