  In this paper we explore the relation between conformal superalgebras with 64
supercharges and maximal supergravity theories in three, four and six
dimensions using twistorial oscillator techniques. The massless fields of N=8
supergravity in four dimensions were shown to fit into a CPT-self-conjugate
doubleton supermultiplet of the conformal superalgebra SU(2,2|8) a long time
ago. We show that the fields of maximal supergravity in three dimensions can
similarly be fitted into the super singleton multiplet of the conformal
superalgebra OSp(16|4,R), which is related to the doubleton supermultiplet of
SU(2,2|8) by dimensional reduction. Moreover, we construct the ultra-short
supermultiplet of the six-dimensional conformal superalgebra OSp(8*|8) and show
that its component fields can be organized in an on-shell superfield. The
ultra-short OSp(8*|8) multiplet reduces to the doubleton supermultiplet of
SU(2,2|8) upon dimensional reduction. We discuss the possibility of a chiral
maximal (4,0) six-dimensional supergravity theory with USp(8) R-symmetry that
reduces to maximal supergravity in four dimensions and is different from
six-dimensional (2,2) maximal supergravity, whose fields cannot be fitted into
a unitary supermultiplet of a simple conformal superalgebra. Such an
interacting theory would be the gravitational analog of the (2,0) theory.
