It is expected that such a theory, while not solving the problem to construct a theory of quantum gravity, would still have a wide range of validity.
From a conceptual viewpoint abandoning the special situation of the Minkowski spacetime could lead to the development of new ideas and tools that turn out to be helpful to understand the concept of a quantum field theory better.
The first point deserves some elaboration: The curved manifolds under consideration are supposed to be solutions to the field equations of General Relativity, i.e. physically realistic spacetimes, so that gravity is modelled classically by the curvature of spacetime. A quantum field theory on such a spacetime should be able to model the situation of elementary particles that feel the effects of gravity while neglecting the effect that the particles themselves have on spacetime (the notion of “particle” is highly nontrivial and problematic in this setting and is to be understood in a metaphorical sense in the given context).
Example: If you let an electron drop from your hand to the ground, that would be a situation that the theory is supposed to describe. While a full theory of quantum gravity still eludes us, a theory of quantum fields on curved spacetimes could be useful as a kind of interpolation. In a certain sense this is already the case, since the laws of black hole thermodynamics were first discovered with the help of this setting.
A classic reference is this:
Recently published review papers:
Romeo Brunetti, Klaus Fredenhagen: Quantum Field Theory on Curved Backgrounds
Robert M. Wald: The Formulation of Quantum Field Theory in Curved Spacetime
Papers about the application of microlocal analysis: