Working towards the aim of axiomatizing realistic quantum field theories in a TQFT-type framework we focus on the simplest class of examples: linear field theories and their perturbation theory. In order to understand quantization it turns out to be useful to introduce an axiomatization of classical field theory also, on manifolds with boundary. We show how geometric quantization together with the Feynman path integral then leads to a quantization functor from (augmented) classical field theories to quantum field theories. We discuss scope, applications, limitations and future directions of this approach.