  We develop a general framework for the quantization of bosonic and fermionic
field theories on affine bundles over arbitrary globally hyperbolic spacetimes.
All concepts and results are formulated using the language of category theory,
which allows us to prove that these models satisfy the principle of general
local covariance. Our analysis is a preparatory step towards a full-fledged
quantization scheme for the Maxwell field, which emphasises the affine bundle
structure of the bundle of principal U(1)-connections. As a by-product, our
construction provides a new class of exactly tractable locally covariant
quantum field theories, which are a mild generalization of the linear ones. We
also show the existence of a functorial assignment of linear quantum field
theories to affine ones. The identification of suitable algebra homomorphisms
enables us to induce whole families of physical states (satisfying the
microlocal spectrum condition) for affine quantum field theories by pulling
back quasi-free Hadamard states of the underlying linear theories.
