In this paper we investigate the thermal-diffusive instability of premixed adiabatic flames in rich hydrogen-air mixtures at normal ambient conditions. Several models which feature the same two-step global kinetics with chain-branching and recombination reaction steps are considered. These global kinetic steps are assumed to be controlled by different elementary reactions. The flame speed and structure are investigated numerically by using the shooting-relaxation algorithms. The stability is studied by means of the Evans function method and by direct integration of the governing partial differential equations. It is demonstrated that the two-step models are capable of accurately predicting the speed and structure of combustion waves as well as flame stability and frequency of pulsations of unsteady combustion waves. (C) 2013 The Combustion Institute. Published by Elsevier Inc. All rights reserved.