||Lossy decode-and-forward (DF) relaying, also known as lossy forwarding (LF), has attracted much attention since it can significantly enhance the transmission reliability and expand the communication coverage at a small increase on computational effort compared to its DF counterpart. Meanwhile, it can further simplify the operations at the relay node by removing the error-detecting operation, e.g., cyclic redundancy check, which is usually encompassed in the conventional DF. Due to these advantages, LF has been intensively investigated with the aim of its applications to various cooperative communication networks having different topologies. This paper offers a comprehensive tutorial on the LF relaying strategy and makes comparisons between LF and DF. As expected, the performance enhancement of LF over DF has been demonstrated in the exemplifying scenario. LF has great potential to be applied in future 5G wireless communication networks, e.g., device- to-device, vehicle-to-vehicle, and machine-to-machine communications.