In stochastic analysis, the k-th signature of a path is a k-way tensor which encodes information about the path in a compact form. When the path varies, the signatures parametrize a subvariety of tensor space, giving a connection to algebraic geometry. In this talk I will present some recent results on varieties on signature tensors, and report on some work in progress (joint with Carlos Amendola, Francesco Galuppi, Angel Rios, and Pierpaola Santarsiero), where we look at these varieties through the lens of representation theory.