Abstract: Brownian motion is continuous random motion, discovered by early 19th Century botanist Robert Brown, studied by Albert Einstein in one of the three 1905 papers that led to his Nobel prize, and finally put on firm mathematical footing by Norbert Wiener in the 1920s. It is intimately tied to local and global geometry, and is an important tool in studying heat flow on more general manifolds. In this talk, I will give an overview of some results on Brownian motion on classical Lie groups, focusing on unitary groups $U_N$ and general linear groups $GL_N$. I will discuss my recent work on the large-N limit of Brownian motions on these groups, their fluctuations, and applications to random matrix theory and operator algebras.