This paper establishes a review of the recent study in the field of hyperspectral (HS) image compression approaches. Recently, image compression techniques have achieved significant advances from diverse types of coding standards/approaches. HS image compression requires an unconventional coding technique because of its unique, multiple-dimensional structure. The data redundancy exists in both inter-band and intra-band methods. The survey summarizes current literature in inter- and intra-band compression methods. The challenges, opportunities, and future research possibilities regarding HS image compression are further discussed. The experimental results are also provided for validity and applicability of the existing HS image compression techniques.