In this paper using a Clifford bundle formalism we examine (a): the strong conditions for existence of conservation laws involving only the energy-momentum and angular momentum of the matter fields on a general Riemann-Cartan spacetime and also in the particular cases of Lorentzian and teleparallel spacetimes and (b): the conditions for the existence of conservation laws of energy-momentum and angular momentum for the matter and gravitational fields when this latter concept can be rigorously defined. We examine in details some misleading and even erroneous and often quoted statements concerning the issues of the conservation laws in General Relativity and Riemann-Cartan (including the particular case of the teleparallel one) theories.
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