The three-way catalytic converter (TWC) used for automobile emission control is the most common reactor for detoxification of automobile exhaust gases. The processes in the TWC include mass transport coupled with simultaneous heterogeneous catalytic oxidation of CO and hydrocarbons, and reduction of NO_x in a monolith – a multichannel flow-through reactor. In the first step, we focus on the analysis of complex dynamical modes stemming from the complexity of the chemical processes involved. Therefore we take the simplest approximation of the system being an isothermal continuous stirred tank reactor (CSTR). The model exhibits various types of dynamics characteristic of nonlinear systems, such as multiple steady states and oscillations found via bifurcation diagrams constructed by numerical continuation techniques. These phenomena are caused by interactions among various reaction and adsorption steps, not by the thermokinetic effect. To account for these dynamical modes, we analyze the reaction mechanism taken from the literature with the use of the stoichiometric network analysis (SNA) – a tool for decomposition of the reaction network and determination of unstable subnetworks. Such reaction subnetworks may possess positive and negative feedback loops, which imply periodic oscillations via a Hopf bifurcation. This analysis is crucial for interpretation of the chemical nature of oscillations observed within certain region of parameters in the oxygen inflow vs. temperature bifurcation diagrams. For example, the analysis of simultaneous CO oxidation and NO_x reduction explains the formation of N₂O – a gas with strong greenhouse effects – predicted by the model in a super-stoichiometric composition of exhaust gases. In a subsequent step, a spatially one-dimensional reaction-diffusion-convection model is examined to find conditions for occurrence of travelling concentration waves, which are due to the excitable nature of the dynamics.