Kinetic theory of polyatomic gases is becoming increasingly important because their non-equilibrium flows are encountered in many applications, such as high-altitude flights and micro-scale systems. In the present talk, I will first give a brief introduction to kinetic theory of polyatomic gases. However, since the Boltzmann equation for a polyatomic gas cannot be written in a concrete form that can be applied to practical problems, its model equations are usually used in applications. I will focus on one of such models, that is, the ellipsoidal statistical (ES) model for a polyatomic gas that was proposed by Andries et al. and explain the form and basic properties of the model. This model is for an ideal polyatomic gas with constant specific heats (thermally perfect gas). I will also extend the ES model to a polyatomic ideal gas with temperature-dependent specific heats (calorically perfect gas). Finally, these models are applied to investigating the problem of shock-wave structure with special interest in CO2 gas, which is known to have a very large bulk viscosity.