The rapid development of medical imaging and the invention of various medicines have benefited mankind and the whole community. Medical image processing is a niche area concerned with the operations and processes of generating images of the human body for clinical purposes. Potential areas such as image acquisition, image enhancement, image compression and storage, and image based visualization also include in medical image processing analysis. Unfortunately, medical image compression dealing with three-dimensional (3-D) modalities still in the pre-matured stage. Along with that, very limited researchers take a challenge to apply hardware on their implementation. Referring to the previous work reviewed, most of the compression method used lossless rather than lossy. For implementation using software, MATLAB and Verilog are the famous candidates among researchers. In term of analysis, most of the previous works conducted objective test compared with subjective test. This paper thoroughly reviews the recent advances in medical image compression mainly in terms of types of compression, software and hardware implementations and performance evaluation. Furthermore, challenges and open research issues are discussed in order to provide perspectives for future potential research. In conclusion, the overall picture of the image processing landscape, where several researchers more focused on software implementations and various combinations of software and hardware implementation.