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We demonstrate that the generators for the local, continuous andinfinitesimal classical gauge symmetry transformations in the cases of (i) theSt$\ddot u$ckelberg-modified massive Abelian 1-form and 2-form theories, and(ii) the massless Abelian 1-form and 2-form free theories owe their origin tothe first-class constraints of the these theories. We establish a connectionbetween the standard forms of the generators and the Noether conserved chargesfor the modified massive and massless versions of the above theories. Wediscuss the appearance of these constraints, within the framework ofBecchi-Rouet-Stora-Tyutin (BRST) formalism, through the physicality criteriaw.r.t. the conserved and nilpotent (anti-)BRST charges. One of the highlightsof our present investigation is the observation that, in the context of themodified massive and massless Abelian 2-form theories, the modified forms ofthe standard Noether (anti-)BRST charges are required which are found to beoff-shell nilpotent and they lead to the appearance of the operator forms ofthe first-class constraints through the physicality criteria at the quantumlevel. We also comment on (i) the existence of the Curci-Ferrari (CF)-typerestrictions on the Abelian 2-form theories (with and without mass), (ii) themodifications in the St$\ddot u$ckelberg-technique for the massive 2D Abelian1-form and 4D Abelian 2-form theories and their consequences, and (iii) theoff-shell nilpotent version of the conserved co-BRST charge and its role in thephysicality criteria for the St$\ddot u$ckelberg-modified 4D massive Abelian2-form theory.

Constraints and Conserved Charges for Modified Massive and Massless Abelian 1-Form and 2-Form Theories: A Brief Review