Collisional-radiative spectral modeling plays a central role in astrophysics, probing phenomena ranging from the chemical evolution of the Universe to the energy production near supermassive black holes in distant quasars. The observed emission lines form in non-equilibrium clouds that have very low densities by laboratory standards, and are powered by energy sources which themselves are not in equilibrium. The spectrum is the result of a large number of microphysical processes, thermal statistics often do not apply, and analytical theory cannot be used. Numerical simulations are used to understand the physical state and the resulting spectrum. The greatest distinction between astrophysical modeling and conventional plasma simulations lies in the range of phenomena that must be considered. A single astronomical object will often have gas with kinetic temperatures of T̃10^6 K, 10^4 K, and T≤ 10^3 K,with the physical state ranging from molecular to fully ionized, and emitting over all wavelengths between the radio and x-ray. Besides atomic, plasma, and chemical physics, condensed matter physics is important because of the presence of small solid `grains' which affect the gas through catalytic reactions and the infrared emission they produce. The ionization, level populations, chemistry, and grain properties must be determined self-consistently, along with the radiation transport, to predict the observed spectrum. Although the challenge is great, so are the rewards. Numerical spectral simulations allow us to read the message contained in the spectrum emitted by objects far from the Earth that existed long ago.