  We investigate the relationship between conformal and spin structure on
lorentzian manifolds and see how their compatibility influences the formulation
of rigid supersymmetric field theories. In dimensions three, four, six and ten,
we show that if the Dirac current associated with a generic spinor defines a
null conformal Killing vector then the spinor must obey a twistor equation with
respect to a certain connection with torsion. Of the theories we consider,
those with classical superconformal symmetry in Minkowski space can be
reformulated as rigid supersymmetric theories on any lorentzian manifold
admitting twistor spinors. In dimensions six and ten, we also describe rigid
supersymmetric gauge theories on bosonic minimally supersymmetric supergravity
backgrounds.
