  A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class
$\mathcal{U}_A(\mathbb{D})$ of Abel universal functions if the family $\{f_r:
0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the space of continuous
functions on $K$, for any proper compact subset $K$ of the unit circle. It has
been recently shown that $\mathcal{U}_A(\mathbb{D})$ is a dense $G_{\delta}$
subset of the space of holomorphic functions on $\mathbb{D}$ endowed with the
topology of local uniform convergence. In this paper, we develop further the
theory of universal radial approximation by investigating the boundary
behaviour of functions in $\mathcal{U}_A(\mathbb{D})$ (local growth, existence
of Picard points and asymptotic values) and the convergence properties of their
Taylor polynomials outside $\mathbb{D}$.
