  Tensors, or multi-linear forms, are important objects in a variety of areas
from analytics, to combinatorics, to computational complexity theory. Notions
of tensor rank aim to quantify the "complexity" of these forms, and are thus
also important. While there is one single definition of rank that completely
captures the complexity of matrices (and thus linear transformations), there is
no definitive analog for tensors. Rather, many notions of tensor rank have been
defined over the years, each with their own set of uses. In this paper we
survey the popular notions of tensor rank. We give a brief history of their
introduction, motivating their existence, and discuss some of their
applications in computer science. We also give proof sketches of recent results
by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between
three key notions of tensor rank over finite fields with at least three
elements.
