Numerical simulations based on the Monte Carlo method offer a powerful approach for detailed studies of complex reaction sequences, such as those associated with heterogeneous catalysis. In this article, we summarize some of the recent work based on discrete models for irreversible surface reactions. Particular emphasis is placed on kinetic phase transitions, bistability, and oscillatory (nonstationary) reactions. In addition to discussing some of the fundamental aspects of nonequilibrium kinetics, we show through specific examples that explicit Monte Carlo simulations can transcend traditional approaches based on rate-equation methods, in particular those invoking the mean-field approximation. This is particularly the case when local correlations and fluctuations among the reactants are important.