A long open problem in probability theory has been the following: Can the graph of planar Brownian motion be split by a straight line? Let $Z_t$ be two-dimensional Brownian motion. Say that a straight line $mathcal L$ is a cut line if there exists a time $t in (0,1)$ such that the trace of ${{ Z_s: 0 leq s < t}}$ lies on one side of $mathcal L$ and the trace of ${{Z_s: t < s < 1}}$ lies on the other side of $mathcal L$. In this volume, the authors provide a solution, discuss related works, and present a number of open problems.