Spectral synthesis techniques for supernovae and kilonovae

Supernovae (SNe) and kilonovae (KNe) are the most violent explosions in cosmos, signalling the destruction of a massive star (core-collapse SN), a white dwarf (thermonuclear SN) and a neutron star (KN), respectively. The ejected debris in these explosions is believed to be the main cosmic source of most elements in the periodic table. However, decoding the spectra of these transients is a challenging task requiring sophisticated spectral synthesis modelling. Here, the techniques for such modelling is reviewed, with particular focus on the computational aspects. We build from a historical review of how methodologies evolved from modelling of stellar winds, to supernovae, to kilonovae, studying various approximations in use for the central physical processes. Similarities and differences in the numeric schemes employed by current codes are discussed, and the path towards improved models is laid out.