In this thesis, I develop a theoretical description of the radiation of an extended, spinning light source on a circular orbit in the symmetry plane of a stationary, axially symmetric and asymptotically flat spacetime. The light source is assumed to be a test particle, in order to neglect its gravitational influence on the background spacetime. I derive the necessary transformations for a reference frame that is at rest on the surface of the rotating emitter, and link the emission angles, relative to the surface of the emitter, to the constants of motion of the light ray. Two emitter geometries are considered: a sphere and a Maclaurin spheroid, which is flattened as a result of its spin. In particular, I apply this theory to an emitter in orbit around a Schwarzschild object, as well as an emitter orbiting a Kerr black hole. In this context, I investigate the influence of the emitter spin on the observables, specifically the polarization plane, redshift and flux, as well as the influence of the black hole rotation. Notably, the position of the emitter orbit where the maximum flux is observed depends on the spin, and non-monotony is observed in the amount of observed flux, varying the spin of the emitter. This theory may find application in describing the emissions of spinning hot spots in accretion disks or neutron stars.