In a process for monitoring the performance of a catalyst (8) for exhaust gases of an internal combustion engine (1) with an exhaust gas system (5, 8, 9), the temperature of two spatially separate cross-sectional regions in the exhaust gas system is measured by temperature sensors (TF1, TF2; TF1', TF2') and monitored by an electronic monitoring device (10). At least one part of the volume of the catalyst which is essential for the catalytic conversion of noxious substances in the exhaust gas is located between the two measurement points. The temperature sensors (TF1, TF2) do not measure discretely but integrally over a representative portion of the cross-sectional region. The temperature measurement points can be integrated in the catalyst (8), which can also preferably be heated electrically (17, 18). In a properly functioning system, the exothermic reaction in the catalytic conversion of noxious substances in the exhaust gas brings about a certain temperature difference between the two measurement points, which can be monitored. The function of the catalyst (8) and/or engine (1) and/or electronic engine control system (2) can be deduced from the sign and/or absolute value and/or temporal variation of the temperature difference. Preferred field of application: motor vehicles with spark ignition engines with an electronic engine control system.