Abstract: A holomorphic function on a planar domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of the Taylor series have the following surprising property: they can approximate arbitrary polynomials on arbitrary compact sets $K$ outside $\Omega$ (provided only that $K$ has connected complement). In the last few years, central questions about universal Taylor series have been addressed using potential theory. In this talk we will discuss some of these results and in particular we will focus on the boundary behaviour of such functions.