Reservoir computing (RC), as a brain-inspired neuromorphic computing algorithm, is capable of fast and energy-efficient temporal data analysis and prediction. Hardware implementation of RC systems can significantly reduce the computing time and energy, but it is hindered by current physical devices. Recently, dynamic memristors have proved to be promising for hardware implementation of such systems, benefiting from their fast and low-energy switching, nonlinear dynamics, and short-term memory behavior. In this work, we review striking results that leverage dynamic memristors to enhance the data processing abilities of RC systems based on resistive switching devices and magnetoresistive devices. The critical characteristic parameters of memristors affecting the performance of RC systems, such as reservoir size and decay time, are identified and discussed. Finally, we summarize the challenges this field faces in reliable and accurate task processing, and forecast the future directions of RC systems.