  Functionals (i.e. functions of functions) are widely used in quantum field
theory and solid-state physics. In this paper, functionals are given a rigorous
mathematical framework and their main properties are described. The choice of
the proper space of test functions (smooth functions) and of the relevant
concept of differential (Bastiani differential) are discussed.
  The relation between the multiple derivatives of a functional and the
corresponding distributions is described in detail. It is proved that, in a
neighborhood of every test function, the support of a smooth functional is
uniformly compactly supported and the order of the corresponding distribution
is uniformly bounded. Relying on a recent work by Yoann Dabrowski, several
spaces of functionals are furnished with a complete and nuclear topology. In
view of physical applications, it is shown that most formal manipulations can
be given a rigorous meaning.
  A new concept of local functionals is proposed and two characterizations of
them 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 a
cohomological approach to quantum field theory is carried out by proving a
global Poincar\'e lemma and defining multi-vector fields and graded functionals
within our framework.
