An increasing number of scientific and enterprise data sets are multidimensional, where data is gathered for every configuration of three or more parameters. For example, physical simulations often track a set of variables in two or three spatial dimensions over time, yielding 4D or 5D data sets. Tensor decompositions are structured representations of multidimensional data that generalize low-rank matrix approximations (2D data). The most common decompositions are called CP and Tucker, but there are many others.  In this talk, I’ll discuss the utility of CP and Tucker decompositions in various applications, how they’re computed numerically, and recently developed techniques for making the computations efficient on parallel computers.