The Discontinuous Reception (DRX) is the most effective timer-based mechanism for User Equipment (UE) power saving. In Long Term Evolution (LTE) systems, the development of the DRX mechanism enormously extends the UE battery life. With the DRX mechanism, a UE is allowed to enter a dormant state. Given a DRX cycle, the UE needs to wake up periodically during the dormancy to check whether it receives new downlink packets or not. The UE can achieve a high sleeping ratio by skipping most channel monitoring occasions. As the mobile network evolved to 5G, the battery life requirement increased to support various new services. 3rd Generation Partnership Project (3GPP) also enhances the DRX mechanism and adds new DRX-related features in the New Radio (NR) Release 16 standard. In addition to the time-based design, 3GPP proposed two signaling-based mechanisms: power saving signal and UE assistance information. This survey paper introduces the latest DRX mechanism in the 3GPP NR standard and summarizes the state-of-the-art research. Researchers have investigated the DRX mechanism in various use cases, such as Web browsing services and heterogeneous networks. They focus on the UE sleep ratio and packet delay and propose corresponding analytical models. New DRX architectures are also discussed to conquer the power-saving problem in specific schemes, especially in the 5G NR networks. This paper categorizes and presents the papers according to the target services and the network scenarios in detail. We also survey the work focusing on the new challenges (such as beamforming and thermal issue) in the NR network and introduce the future research directions in the 6G era.