Let f be a holomorphic function on the open unit disc D, and let $S_n$ denote the $n$-th partial sum of the Taylor series of f centred at 0. We say that f is universal if, outside D, every holomorphic function can be approximated (in a sense to be made precise) by a subsequence of $(S_n)$. Such universal functions f
were shown to exist by Nestoridis in 1996, though milder versions have
been known about since 1970. We discuss the construction and properties
of Nestoridis' universal functions and mention, in particular, recent
joint work with George Costakis concerning their radial limiting
behaviour near the boundary of D.