  We show how the rigid conformal supersymmetries associated with a certain
class of pseudo-Riemannian spin manifolds define a Lie superalgebra. The even
part of this superalgebra contains conformal isometries and constant
R-symmetries. The odd part is generated by twistor spinors valued in a
particular R-symmetry representation. We prove that any manifold which admits a
conformal symmetry superalgebra of this type must generically have dimension
less than seven. Moreover, in dimensions three, four, five and six, we provide
the generic data from which the conformal symmetry superalgebra is prescribed.
For conformally flat metrics in these dimensions, and compact R-symmetry, we
identify each of the associated conformal symmetry superalgebras with one of
the conformal superalgebras classified by Nahm. We also describe several
examples for Lorentzian metrics that are not conformally flat.
