  The standard Black Holes (BHs) in General Relativity, as well as other
ultra-compact objects (with or without an event horizon) admit planar circular
photon orbits. These light rings (LRs) determine several spacetime properties.
For instance, stable LRs trigger instabilities and, in spherical symmetry,
(unstable) LRs completely determine BH shadows. In generic stationary,
axi-symmetric spacetimes, non-planar bound photon orbits may also exist,
regardless of the integrability properties of the photon motion. We suggest a
classification of these fundamental photon orbits (FPOs) and, using Poincar\'e
maps, determine a criterion for their stability. For the Kerr BH, all FPOs are
unstable (similarly to its LRs) and completely determine the Kerr shadow. But
in non-Kerr spacetimes, stable FPOs may also exist, even when all LRs are
unstable, triggering new instabilities. We illustrate this for the case of Kerr
BHs with Proca hair, wherein, moreover, qualitatively novel shadows with a
cuspy edge exist, a feature that can be understood from the interplay between
stable and unstable FPOs. FPOs are the natural generalisation of LRs beyond
spherical symmetry and should generalise the LRs key role in different
spacetime properties.
