A crucial aspect of quality control for Additive Manufacturing (AM) processes is the acquisition of diverse data from the entire lifecycle of the product. AM data has grown significantly in terms of diversity and volumes, resulting in diverse data formats of increasing volumes, including time series, images, and point clouds. Large quantities of these data are essential for effective in-situ process monitoring and ex-situ non-destructive evaluation. However, this will result in large manufacturing and inspection datasets that are difficult to manage for users, which will delay the broader adoption of AM for mission critical applications. This motivates the urgent need for effective big data management for AM processes. Data compression is a core strategy that reduces big data volumes by restructuring original data to their compressed representations. This paper first summarizes all AM data that is generated during the AM product lifecycle. This work then reviews the state-of-the-art data compression methods that either have been implemented or can be implemented for compressing AM data. Lastly, some challenges and opportunities for AM data compression are presented for future research.