Array DBMSs strive to be the best systems for managing, processing, and even visualizing big N-d arrays. The last decade blossomed with R&D in array DBMS, making it a young and fast-evolving area. We present the first comprehensive tutorial on array DBMS R&D. We start from past impactful results that are still relevant today, then we cover contemporary array DBMSs, array-oriented systems, and state-of-the-art research in array management, flavored with numerous promising R&D opportunities for future work. A great deal of our tutorial was not covered in any previous tutorial or survey article. Advanced array management research is just emerging and many R&D opportunities still “lie on the surface”. Hence, nowadays we have the most favorable conditions to start contributing to this research area. This tutorial will jump-start such efforts.