  Motivated by DeWitt's viewpoint of covariant field theory, we define a
general notion of non-local classical observable that applies to many physical
lagrangian systems (with bosonic and fermionic variables), by using methods
that are now standard in algebraic geometry. We review the (standard) methods
of local functional calculus, as they are presented by Beilinson and Drinfeld,
and relate them to our construction. We partially explain the relation of these
with the Vinogradov's secondary calculus. The methods present here are all
necessary to understand mathematically properly and with simple notions the
full renormalization of the standard model, based on functional integral
methods. This article can be seen as an introduction to well grounded classical
physical mathematics, and as a good starting point to study quantum physical
mathematics, that make frequent use of non-local functionals, like for example
in the computation of Wilson's effective action. We finish by describing
briefly a coordinate free approach to the classical Batalin-Vilkovisky
formalism for general gauge theories, in the language of homotopical geometry.
