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Functionals (i.e. functions of functions) are widely used in quantum fieldtheory and solid-state physics. In this paper, functionals are given a rigorousmathematical framework and their main properties are described. The choice ofthe proper space of test functions (smooth functions) and of the relevantconcept of differential (Bastiani differential) are discussed.  The relation between the multiple derivatives of a functional and thecorresponding distributions is described in detail. It is proved that, in aneighborhood of every test function, the support of a smooth functional isuniformly compactly supported and the order of the corresponding distributionis uniformly bounded. Relying on a recent work by Yoann Dabrowski, severalspaces of functionals are furnished with a complete and nuclear topology. Inview of physical applications, it is shown that most formal manipulations canbe given a rigorous meaning.  A new concept of local functionals is proposed and two characterizations ofthem are given: the first one uses the additivity (or Hammerstein) property,the second one is a variant of Peetre's theorem. Finally, the first step of acohomological approach to quantum field theory is carried out by proving aglobal Poincar\'e lemma and defining multi-vector fields and graded functionalswithin our framework.

Properties of field functionals and characterization of local functionals