The INARCH(1) model has been proposed in the literature as a simple, but practically relevant, two-parameter model for processes of overdispersed counts with an autoregressive serial dependence structure. In this research, we develop approaches for monitoring INARCH(1) processes for detecting shifts in the process parameters. Several cumulative sum control charts are derived directly from the log-likelihood ratios for various types of shifts in the INARCH(1) model parameters. We define zero-state (worst-state) and steady-state average run length metrics and discuss their computation for the proposed charts. An extensive study indicates that these charts perform well in detecting changes in the process. A real-data example of strike counts is used to illustrate process monitoring.