A goal of runtime software-fault monitoring is to observe software behavior to determine whether it complies with its intended behavior. Monitoring allows one to analyze and recover from detected faults, providing additional defense against catastrophic failure. Although runtime monitoring has been in use for over 30 years, there is renewed interest in its application to fault detection and recovery, largely because of the increasing complexity and ubiquitous nature of software systems. We present taxonomy that developers and researchers can use to analyze and differentiate recent developments in runtime software fault-monitoring approaches. The taxonomy categorizes the various runtime monitoring research by classifying the elements that are considered essential for building a monitoring system, i.e., the specification language used to define properties; the monitoring mechanism that oversees the program's execution; and the event handler that captures and communicates monitoring results. After describing the taxonomy, the paper presents the classification of the software-fault monitoring systems described in the literature.