Noether's theorem states that for a Lagrangian field theory, symmetries of the action gives rise to conserved currents and charges. The most well-known symmetries are those which arise from Killing symmetries of the background spacetime, eg. the Poincaré symmetries of Minkowski space. However, for fields with spin, even on Minkowski space, there are non-trivial conservation laws which do not arise in this manner. These include conservation laws associated to Killing tensors, eg. Lipkin's zilches, as well as conservation laws related to duality symmetry, eg.. helicity, intrinsic spin and orbital angular momentum, which are relevant in considering the spin Hall effect. For fields on Kerr, analogous constructions yield non-trivial conservation laws associated to the Carter constant, for the scalar field as well as for spinning fields including Maxwell and linearized gravity. In this talk, I will give some background on these non-classical conservation laws, and give examples of the underlying symmetries. I will focus on case of Maxwell on Minkowski, but will also mention some generalizations to fields of higher spin and to fields on the Kerr spacetime.