Rapid advancement in the development of hyperspectral image analysis techniques has led to specialized hyperspectral missions. It results in the bulk transmission of hyperspectral images from sensors to analysis centers and finally to data centers. Storage of these large size images is a critical issue that is handled by compression techniques. This survey focuses on different hyperspectral image compression algorithms that have been classified into two broad categories based on eight internal and six external parameters. In addition, we identified research challenges and suggested future scope for each technique. The detailed classification used in this paper can categorize other compression algorithms and may help in selecting research objectives.