  In the framework of classical field theory, we first review the Noether
theory of symmetries, with simple rederivations of its essential results, with
special emphasis given to the Noether identities for gauge theories. Will this
baggage on board, we next discuss in detail, for Poincar\'e invariant theories
in flat spacetime, the differences between the Belinfante energy-momentum
tensor and a family of Hilbert energy-momentum tensors. All these tensors
coincide on shell but they split their duties in the following sense:
Belinfante's tensor is the one to use in order to obtain the generators of
Poincar\'e symmetries and it is a basic ingredient of the generators of other
eventual spacetime symmetries which may happen to exist. Instead, Hilbert
tensors are the means to test whether a theory contains other spacetime
symmetries beyond Poincar\'e. We discuss at length the case of scale and
conformal symmetry, of which we give some examples. We show, for Poincar\'e
invariant Lagrangians, that the realization of scale invariance selects a
unique Hilbert tensor which allows for an easy test as to whether conformal
invariance is also realized. Finally we make some basic remarks on metric
generally covariant theories and classical field theory in a fixed curved
bakground.
