  We give an overview of some conceptual difficulties, sometimes called
paradoxes, that have puzzled for years the physical interpetation of classical
canonical gravity and, by extension, the canonical formulation of generally
covariant theories. We identify these difficulties as stemming form some
terminological misunderstandings as to what is meant by "gauge invariance", or
what is understood classically by a "physical state". We make a thorough
analysis of the issue and show that all purported paradoxes disappear when the
right terminology is in place. Since this issue is connected with the search of
observables - gauge invariant quantities - for these theories, we formally show
that time evolving observables can be constructed for every observer. This
construction relies on the fixation of the gauge freedom of diffeomorphism
invariance by means of a scalar coordinatization. We stress the condition that
the coordinatization must be made with scalars. As an example of our method for
obtaining observables we discuss the case of the massive particle in AdS
spacetime.
