We investigate the radial behavior of holomorphic functions in the unit ball $B$ of $\mtc^n$. In particular, we prove the existence of universal holomorphic functions $f$ in the following sense : given any measurable function $\vphi$ on $\partial B$, there is a sequence $(r_n)_{n\geq 1}$, $0<r_n<1$, that converges to 1, such that $f(r_n\xi)$ converges to $\vphi(\xi)$ for almost every $\xi\,{\in}\,\partial B$.