  George Mackey suggested in 1975 that there should be analogies between the
irreducible unitary representations of a noncompact reductive Lie group $G$ and
those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal
compact subgroup of $G$ and a vector space. He conjectured the existence of a
natural one-to-one correspondence between "most" irreducible (tempered)
representations of $G$ and "most" irreducible (unitary) representations of
$G_0$. We here describe a simple and natural bijection between the tempered
duals of both groups, and an extension to a one-to-one correspondence between
the admissible duals.
