Models are developed for decision making about monitoring and maintenance of systems whose performance through time is described by a general stochastic process. The system is monitored and preventive and corrective maintenance actions are carried out in response to the observed system state. The decision process is simplified by using the maximum process as a decision variable. The models developed generalize age replacement models and other simple maintenance strategies. The approach can deal with failures that prevent the system functioning further, and also failures that are defined by regulation or economic considerations. Attention is restricted to perfect repair and inspection, but the structure provides the framework for further developments.