  We study the Burau representation of the braid group $B_n$ in the case where
$n=3$. We give three novel topological proofs that the Burau representation of
$B_3$ is faithful, and a proof that it's faithful modulo $p$ for all integers
$p>1$. We then classify conjugacy classes in the image of the Burau
representation in $\text{GL}(2, \mathbb{Z}[t, t^{-1}])$ in a way that takes
account of the fact that braids are geometrically oriented, and use that fact
to give a new, linear time solution to the conjugacy problem in $B_3$. We pose
several open problems about $B_n$.
