Event-triggered fault diagnosis has attracted tremendous research attention in the last decade due to its superiority in improving the utilisation efficiency of communication resources. Different from traditional works of time-driven, event-triggered schemes are used to determine whether the current measurement output should be released to the fault detection filter, while the sensor data not satisfying a predefined triggering condition will be discarded directly. As such, research on event-triggered fault diagnosis has been a challenging issue and many outstanding results have been reported. This paper presents a survey of model-based event-triggered fault detection (FD) and fault estimation (FE) methods mainly based on the techniques of residual generation. First, an overview of recent advances in state estimation-based methods of event-triggered FD is provided, which include the event-triggered FD for dynamic systems subject to Gaussian noises, the H-infinity filtering formulation of event-triggered FD, and the event-triggered H-i/H-infinity optimisation-based FD. Second, the representative results of parity space-based event-triggered FD are reviewed. Third, recent results on event-triggered FE are also reviewed. Finally, several challenging issues on event-triggered fault diagnosis are provided for future research.