  This survey is devoted to Martin Hairer's Reconstruction Theorem, which is
one of the cornerstones of his theory of Regularity Structures. Our aim is to
give a new self-contained and elementary proof of this Theorem, together with
some applications, including a characterization, based on a single arbitrary
test function, of negative H\"older spaces. We present the Reconstruction
Theorem as a general result in the theory of distributions that can be
understood without any knowledge of Regularity Structures themselves, which we
do not even need to define.
