A holomorphic function $f$ on a simply connected domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $\Omega$ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $\Omega$ is the unit disc, that such functions have extreme angular boundary behaviour.

A generalized approach to several universality results is given by replacing holomorphic or harmonic functions by zero solutions of arbitrary linear partial differential operators. Instead of the approximation theorems of Runge and others, we use an approximation theorem of Hörmander.