The rank of a tensor T in V_1 \otimes V_2 ... \otimes V_n is the smallest integer k such that T can be written as a sum of k pure tensors. This notion coincides with matrix rank if there are only 2 tensor factors. However, the situation is considerably more complicated when there are more tensor factors. In this talk I will attempt to describe the importance as well as the challenges in understanding tensor rank and other related notions. Speaker(s): Viswambhara Makam (University of Michigan)