The Cellular Internet of Things (CIoT), a new paradigm, paves the way for a large-scale deployment of IoT devices. CIoT promises enhanced coverage and massive deployment of low-cost IoT devices with an expected battery life of up to 10 years. However, such a long battery life can only be achieved provided the CIoT device is configured with energy efficiency in mind. This paper conducts a comprehensive survey on energy-saving solutions in 3GPP-based CIoT networks. In comparison to current studies, the contribution of this paper is the classification and an extensive analysis of existing energy-saving solutions for CIoT, e.g., configuration of particular parameter values and software modifications of transport- or radio-layer protocols, while also stressing key parameters impacting the energy consumption such as the frequency of data reporting, discontinuous reception cycles (DRX), and Radio Resource Control (RRC) timers. In addition, we discuss shortcomings, limitations, and possible opportunities which can be investigated in the future to reduce the energy consumption of CIoT devices.