This talk will focus on developments in the theory of the ranks of tensors. We will begin by recalling the origins of some of these notions, as well as some of their current applications. After that, we will review some of the major advances in this area‘s accelerating development that have been witnessed by the last ten years. Thereafter, we will focus specifically on the extension in some form of basic properties of the rank of matrices to the ranks of tensors. Finally, we will conclude by outlining several directions of research aimed at addressing central difficulties behind several open questions regarding these ranks.