The rapid growth of remote sensing big data (RSBD) has attracted considerable attention from both academia and industry. Despite the progress of computer technologies, conventional computing implementations have become technically inefficient for processing RSBD. Cloud computing is effective in activating and mining large-scale heterogeneous data and has been widely applied to RSBD over the past years. This study performs a technical review of cloud-based RSBD storage and computing from an interdisciplinary viewpoint of remote sensing and computer science. First, we elaborate on four critical technical challenges resulting from the scale expansion of RSBD applications, i.e. raster storage, metadata management, data homogeneity, and computing paradigms. Second, we introduce state-of-the-art cloud-based data management technologies for RSBD storage. The unit for manipulating remote sensing data has evolved due to the scale expansion and use of novel technologies, which we name the RSBD data model. Four data models are suggested, i.e. scenes, ARD, data cubes, and composite layers. Third, we summarize recent research on the application of various cloud-based parallel computing technologies to RSBD computing implementations. Finally, we categorize the architectures of mainstream RSBD platforms. This research provides a comprehensive review of the fundamental issues of RSBD for computing experts and remote sensing researchers.