Event processing is being used extensively in diverse application domains. Simple and composite events play a critical role in event processing systems and were identified based on application domains. They were formally defined using detection-based (point-based) and occurrence-based (interval-based) semantics over various consumption modes. Even though both the semantics are required they are insufficient for handling emerging applications such as information security, stream and sensor data processing systems. Generalizing the event specification and detection is inevitable for supporting these new applications that were not foreseen by extant systems. First, we motivate the need for generalization using applications from diverse domains. Second, we generalize and formalize primitive and composite events. Finally, we briefly discuss how generalized events can be detected using event registrar graphs.