We give a comprehensive review of various methods to define currents and theenergy-momentum tensor in classical field theory, with emphasis on a geometricpoint of view. The necessity of ``improving'' the expressions provided by thecanonical Noether procedure is addressed and given an adequate geometricframework. The main new ingredient is the explicit formulation of a principleof ``ultralocality'' with respect to the symmetry generators, which is shown tofix the ambiguity inherent in the procedure of improvement and guide it towardsa unique answer: when combined with the appropriate splitting of the fieldsinto sectors, it leads to the well-known expressions for the current as thevariational derivative of the matter field Lagrangian with respect to the gaugefield and for the energy-momentum tensor as the variational derivative of thematter field Lagrangian with respect to the metric tensor. In the second case,the procedure is shown to work even when the matter field Lagrangian dependsexplicitly on the curvature, thus establishing the correct relation betweenscale invariance, in the form of local Weyl invariance ``on shell'', andtracelessness of the energy-momentum tensor, required for a consistentdefinition of the concept of a conformal field theory.
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