  In the first part of this thesis we study the generalization of the recent
algebraic approach to classical field theory by proposing a more general
setting based on the manifold of smooth sections of a non-trivial fiber bundle.
Central is the notion of observables/functionals over such sections,
\textit{i.e.} appropriate smooth functions on them. The kinematic will be
further specified by means of Peierls brackets, which in turn are defined via
the causal propagators of linearized field equations. In the second part we
implement deformation quantization of the algebras obtained above in the
simpler setting of scalar field theory. Wick powers and time ordered products
for quantum field theories in curved spacetimes are defined by giving a set of
axioms which, when implemented, defines uniquely, up to some classifiable
ambiguities, the aforementioned quantities. Those ambiguities are known to be
tightly restrained by locality, covariance and other regularity conditions. One
of the additional constraints used was to require continuous and analytic
dependence on the metric and coupling parameters. It was recently shown that
this rather strong requirement could be weakened, in the case of Wick powers,
to the so-called parametrized microlocal spectrum condition. We therefore show
the existence of Wick powers satisfying the above condition and extend this
axiom to time ordered products, while reestablishing the usual uniqueness and
existence results in light of the new constraint.
