  We study the minimal unitary representations of non-compact groups and
supergroups obtained by quantization of their geometric realizations as
quasi-conformal groups and supergroups. The quasi-conformal groups G leave
generalized light-cones, defined by a quartic norm, invariant and have maximal
rank subgroups of the form H X SL(2,R) such that G/H X SL(2,R) are
para-quaternionic symmetric spaces. We give a unified formulation of the
minimal unitary representations of simple non-compact groups of type A_2, G_2,
D_4, F_4, E_6, E_7, E_8 and Sp(2n,R). The minimal UIRs of Sp(2n,R) are simply
the singleton representations and correspond to a degenerate limit of the
unified construction. The minimal unitary representations of the other
noncompact groups SU(m,n), SO(m,n), SO*(2n) and SL(m,R) are also given
explicitly.
  We extend our formalism to define and construct the corresponding minimal
representations of non-compact supergroups G whose even subgroups are of the
form H X SL(2,R). If H is noncompact then the supergroup G does not admit any
unitary representations, in general. The unified construction with H simple or
Abelian leads to the minimal representations of G(3), F(4) and OSp(n|2,R) (in
the degenerate limit). The minimal unitary representations of OSp(n|2,R) with
the even subgroup SO(n) X SL(2,R) are the singleton representations. We also
give the minimal realization of one parameter family of Lie superalgebras
D(2,1;\sigma).
