  The signature of a parametric curve is a sequence of tensors whose entries
are iterated integrals. This construction is central to the theory of rough
paths in stochastic analysis. It is here examined through the lens of algebraic
geometry. We introduce varieties of signature tensors for both deterministic
paths and random paths. For the former, we focus on piecewise linear paths, on
polynomial paths, and on varieties derived from free nilpotent Lie groups. For
the latter, we focus on Brownian motion and its mixtures.
