**Abstract:** This talk is intended for those who, like the speaker, have at some point wondered whether there is a theory of three- or higher-dimensional matrices that parallels matrix theory. We will explain why a d-dimensional hypermatrix is related to but not quite the same an order-d tensor. We discuss how notions like rank, norm, determinant, eigen and singular values may be generalized to hypermatrices. We will see that, far from being artificial constructs, these notions have appeared naturally in a wide range of applications and can be enormously useful. We will examine several examples, highlighting three from the speaker's recent work: (i) rank of 3-hypermatrices and blind source separation in signal processing, (ii) positive definiteness of 6-hypermatrices and self-concordance in convex optimization, (iii) nuclear norm of 3-hypermatrices and bipartite separability in quantum computing. [(i) is joint work with Pierre Comon and (iii) is joint work with Shmuel Friedland.]