Mobile communication has a great potential to the users due to fulfilling the dreams of real-time multimedia communication like voice, image, and text. The huge amount of data redundancy in still image should be compressed using exact image compression algorithm (ICA) before transmitting via wireless channel. Thus, an ICA should be adaptive, simple, and cost-effective and suitable for feasible implementation. Hardware implementation of the different algorithms has improved using modern, fast, and cost-effective technologies. The main aim of this paper is to review and demonstrate various ICAs developed based on image transmission via wireless channel as well as their hardware implementation. Finally, this review makes bridge for researchers to the future relative studies between different algorithms and architectures, and stands as a reference point for developing more controlling and flexible structures.