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We discuss aspects of global and gauged symmetries in quantum field theoryand quantum gravity, focusing on discrete gauge symmetries. An effectiveLagrangian description of $\Z_p$ gauge theories shows that they are associatedwith an emergent $\Z_p$ one-form (Kalb-Ramond) gauge symmetry. Thisunderstanding leads us to uncover new observables and new phenomena innonlinear $\sigma$-models. It also allows us to expand on Polchinski'sclassification of cosmic strings. We argue that in models of quantum gravity,there are no global symmetries, all continuous gauge symmetries are compact,and all charges allowed by Dirac quantization are present in the spectrum.These conjectures are not new, but we present them from a streamlined andunified perspective. Finally, our discussion about string charges andsymmetries leads to a more physical and more complete understanding of recentlyfound consistency conditions of supergravity.

Symmetries and Strings in Field Theory and Gravity