The quantum theory of rotation angles [S.M. Barnett and D.T. Pegg, Phys. Rev. A 41 3427 (1990)] is generalized to non-integer values of the orbital angular momentum. This requires the introduction of an additional parameter, the orientation of a phase discontinuity associated with fractional values of the orbital angular momentum. We apply our formalism to the propagation of light modes with fractional orbital angular momentum in the paraxial and non-paraxial regime.