  These lecture notes are intended as an introduction to several notions of
tensor rank and their connections to the asymptotic complexity of matrix
multiplication. The latter is studied with the exponent of matrix
multiplication, which will be expressed in terms of tensor (border) rank,
(border) symmetric rank and the asymptotic rank of certain tensors. We
introduce the multilinear rank of a tensor as well, deal with the concept of
tensor equivalence and study prehomogeneous vector spaces with the castling
transform. Moreover, we treat Apolarity Theory and use it to determine the
symmetric rank (Waring rank) of some symmetric tensors.
