We investigate theoretically shear banding in large amplitude oscillatory shear (LAOS) of polymeric and wormlike micellar surfactant fluids. In large amplitude oscillatory shear strain, we observe banding at low frequencies and sufficiently high strain rate amplitudes in fluids for which the underlying stationary constitutive curve of shear stress as a function of shear rate is nonmonotonic. This is the direct (and relatively trivial) analog of quasisteady state banding seen in slow strain rate sweeps along the flow curve. At higher frequencies and sufficiently high strain amplitudes, we report a different but related phenomenon, which we call "elastic" shear banding. This is associated with an overshoot in the elastic (Lissajous-Bowditch) curve of stress as a function of strain and we suggest that it might arise rather widely even in fluids that have a monotonic underlying constitutive curve, and so do not show steady state banding if under a steadily applied shear flow. It is analogous to the elastic banding triggered by stress overshoot in a fast shear startup predicted previously by R. L. Moorcroft and S. M. Fielding [Phys. Rev. Lett. 110(8), 086001 (2013)], but could be more readily observable experimentally in this oscillatory protocol due to its recurrence in each half cycle. In large amplitude oscillatory shear stress, we report shear banding in fluids that shear thin strongly enough to have either a negatively or a weakly positively sloping region in the underlying constitutive curve, noting again that fluids in the latter category do not display steady state banding in a steadily applied flow. This banding is triggered in each half cycle as the stress magnitude transits the region of weak slope in an upward direction such that the fluid effectively yields. It is strongly reminiscent of the transient banding predicted previously in step stress [R. L. Moorcroft and S. M. Fielding, Phys. Rev. Lett. 110(8), 086001 (2013)]. Our numerical calculations are performed in the Rolie-poly model of polymers And wormlike micelles, but we also provide arguments suggesting that our results should apply more widely. Besides banding in the shear strain rate profile, which can be measured by velocimetry, we also predict banding in the shear and normal stress components, measurable by birefringence. As a backdrop to understanding the new results on shear banding in LAOS, we also briefly review earlier work on banding in other time-dependent protocols, focusing in particular on shear startup and step stress. (C) 2016 The Society of Rheology.