i will discuss four families of random matrices. the first two are classical: a gaussian measure on the space of $n\times n$ hermitian matrices (\textquotedblleft gaussian unitary ensemble\textquotedblright) and a gaussian measure on the space of all $n\times n$ complex matrices (\textquotedblleft ginibre ensemble\textquotedblright). as $n\rightarrow\infty,$ the eigenvalues of the gaussian unitary ensemble concentrate onto an interval with a semicircular density, while the eigenvalues of the ginibre ensemble become uniformly distributed in a disk in the complex plane. now, the space of $n\times n$ hermitian matrices can be identified with the lie algebra $u(n)$ of the unitary group $u(n),$ and the gaussian unitary ensemble is the distribution of brownian motion in $u(n).$ similarly, the space of all $n\times n$ matrices is the lie algebra $gl(n;\mathbb{c})$ of the general linear group $gl(n;\mathbb{c})$ and the ginibre ensemble is the distribution of brownian motion in $gl(n;\mathbb{c}).$ it is then natural to consider also brownian motions in the groups $u(n)$ and $gl(n;\mathbb{c})$ themselves. the eigenvalues for brownian motion in $u(n)$ have a known limiting distribution in the unit circle. the eigenvalues for brownian motion in $gl(n;\mathbb{c})$ have received little attention up to now. assuming that the eigenvalues have a limiting distribution, recent results of mine with kemp show that the limiting distribution is supported in a certain domain $\sigma_{t}$ in the complex plane. the figure shows the domain for $t=3.85$, along with a plot of the eigenvalues for $n=2,000.$ one notably feature of the domains is that they change topology from simply connected to doubly connected at $t=4.$ i will give background on all four families of random matrices, describe our new results, and mention some ideas in the proof.