Topological expansion is a method of representing a random matrix computation as a sum over topological gluings. This method is particularly useful in the multi-matrix setting in which we consider expressions in independent and in-general-position matrices. We will discuss topological expansion for $\beta$-matrices, where $\beta$ is the Dyson parameter interpolating between the real case $\beta=1$, the complex case $\beta=2$, and the quaternionic case $\beta=4$.