  The signature of Brownian motion in $\mathbb{R}^{d}$ over a running time
interval $[0,T]$ is the collection of all iterated Stratonovich path integrals
along the Brownian motion. We show that, in dimension $d\geq 2$, almost all
Brownian motion sample paths (running up to time $T$) are determined by its
signature over $[0,T]$
