Manipulating radiation is important for a variety of optoelectronic applications, such as on-chip lasers, energy-efficient grating couplers, and antennas for light detection and ranging. Although designing and optimizing those optoelectronic devices are usually believed to be an engineering-oriented task, recent research reveals that the principles underlying radiation manipulation are closely connected to the concept of topology—the study of properties that are invariant under continuous deformations. In this review, we summarize a series of advances of the physics, phenomena, and applications related to radiation manipulation, in which topological concepts were adopted. Radiation could carry energy escaping from the system, breaking the energy conservation. The non-Hermiticity of such systems brings quite different physical consequences when comparing with the Hermitian counterparts and, hence, also results in the emergence of many interesting and extraordinary phenomena. In particular, it is found that the perfect trapping of light can still be realized in such non-Hermitian systems because of the photonic realization of bound states in the continuum. The fundamental nature of bound states in the continuum has been identified to be topological: they are essentially topological defects of the polarization vector field in momentum space, depicted by a kind of topological invariant named topological charges. Therefore, manipulation of radiation channels can be realized by controlling the topological charge evolution in momentum space. It is also demonstrated that the photonic states accompanied with different topological charges generate vortex beams with unique far-field radiation patterns, and ultra-fast switching of such vortex beams is demonstrated according to this principle. The progresses of topological photonics upon light radiation show that the topology is not just mathematical convenience for depicting photonic systems, but has brought realistic consequences in manipulating light and will boost the applications of photonics and optoelectronics in many aspects.