  By conformally coupling vector and hyper multiplets in Minkowski space, we
obtain a class of field theories with extended rigid conformal supersymmetry on
any Lorentzian four-manifold admitting twistor spinors. We construct the
conformal symmetry superalgebras which describe classical symmetries of these
theories and derive an appropriate BRST operator in curved spacetime. In the
process, we elucidate the general framework of cohomological algebra which
underpins the construction. We then consider the corresponding perturbative
quantum field theories. In particular, we examine the conditions necessary for
conformal supersymmetries to be preserved at the quantum level, i.e. when the
BRST operator commutes with the perturbatively defined S-matrix, which ensures
superconformal invariance of amplitudes. To this end, we prescribe a
renormalization scheme for time-ordered products that enter the perturbative
S-matrix and show that such products obey certain Ward identities in curved
spacetime. These identities allow us to recast the problem in terms of the
cohomology of the BRST operator. Through a careful analysis of this cohomology,
and of the renormalization group in curved spacetime, we establish precise
criteria which ensure that all conformal supersymmetries are preserved at the
quantum level. As a by-product, we provide a rigorous proof that the
beta-function for such theories is one-loop exact. We also briefly discuss the
construction of chiral rings and the role of non-perturbative effects in curved
spacetime.
