We revisit internal energy relaxation processes and related bulk viscosity coefficients in fluid models derived from the kinetic theory. We discuss the apparition of bulk viscosity coefficients in relaxation regimes and the links with equilibrium one-temperature bulk viscosity coefficients. Multiple temperature models of single species fluids are investigated as well as state-to-state models for mixtures of gases. Monte Carlo numerical simulations of internal energy relaxation processes in polyatomic gases are shown to fully agree with the theoretical results. The impact of bulk viscosity in fluid mechanics is also addressed as well as various mathematical aspects of internal energy relaxation and Chapman-Enskog asymptotic expansion for a two-temperature fluid model.