Lossy decode-and-forward (DF) relaying, also referred to as lossy forwarding (LF), can significantly enhance the transmission reliability and expand the communication coverage at the cost of a small increase in computational effort compared to its DF counterpart. Furthermore, it can further simplify the operations at the relay nodes by removing the error-detecting operation, e.g., cyclic redundancy check, which is used in the conventional DF systems. Due to these advantages, LF has been intensively investigated with the aim of its applications to various cooperative communication networks with different topologies. This paper offers a comprehensive literature review on the LF relaying strategy and makes comparisons between LF and DF. Five basic exemplifying scenarios are taken into consideration. These are the three-node network, the single-source multi-relay network with direct source-to-destination link, the multiple access relay channel, the two-way relay network, and the general multi-source multi-relay network. The paper includes not only theoretical performance limit analyses, but also performance evaluation by employing low-complexity accumulator aided turbo codes at the sources and relays as well as joint decoding at the destination. As expected, the performance enhancement in terms of outage probability, frame error rate, and $ {\epsilon }$ -outage achievable rate by LF over DF is significant, which is demonstrated in all the exemplifying scenarios in the literatures. Hence LF has a great potential to be applied to future 5G wireless communication networks, e.g., device-to-device, vehicle-to-vehicle, and machine-type communications, which are composed of the aforementioned exemplifying scenarios.