  Let $0<r<1/4$, and $f$ be a non-vanishing continuous function in $|z|\leq r$,
that is analytic in the interior. Voronin's universality theorem asserts that
translates of the Riemann zeta function $\zeta(3/4 + z + it)$ can approximate
$f$ uniformly in $|z| < r$ to any given precision $\varepsilon$, and moreover
that the set of such $t \in [0, T]$ has measure at least $c(\varepsilon) T$ for
some $c(\varepsilon) > 0$, once $T$ is large enough. This was refined by Bagchi
who showed that the measure of such $t \in [0,T]$ is $(c(\varepsilon) + o(1))
T$, for all but at most countably many $\varepsilon > 0$. Using a completely
different approach, we obtain the first effective version of Voronin's Theorem,
by showing that in the rate of convergence one can save a small power of the
logarithm of $T$. Our method is flexible, and can be generalized to other
$L$-functions in the $t$-aspect, as well as to families of $L$-functions in the
conductor aspect.
