In today’s data networks, the main protocol used to ensure reliable communications is the transmission control protocol (TCP). The TCP performance is largely determined by the used congestion control (CC) algorithm. TCP CC algorithms have evolved over the past three decades and a large number of CC algorithm variations have been developed to accommodate various network environments. The fifth-generation (5G) mobile network presents a new challenge for the implementation of the TCP CC mechanism, since networks will operate in environments with huge user device density and vast traffic flows. In contrast to the pre-5G networks that operate in the sub-6 GHz bands, the implementation of TCP CC algorithms in 5G mmWave communications will be further compromised with high variations in channel quality and susceptibility to blockages due to high penetration losses and atmospheric absorptions. These challenges will be particularly present in environments such as sensor networks and Internet of Things (IoT) applications. To alleviate these challenges, this paper provides an overview of the most popular single-flow and multy-flow TCP CC algorithms used in pre-5G networks. The related work on the previous examinations of TCP CC algorithm performance in 5G networks is further presented. A possible implementation of TCP CC algorithms is thoroughly analysed with respect to the specificities of 5G networks, such as the usage of high frequencies in the mmWave spectrum, the frequent horizontal and vertical handovers, the implementation of the 5G core network, the usage of beamforming and data buffering, the exploitation of edge computing, and the constantly transmitted always-on signals. Moreover, the capabilities of machine learning technique implementations for the improvement of TCPs CC performance have been presented last, with a discussion on future research opportunities that can contribute to the improvement of TCP CC implementation in 5G networks. This survey paper can serve as the basis for the development of novel solutions that will ensure the reliable implementation of TCP CC in different usage scenarios of 5G networks.