  It is shown that every holomorphic map $f$ from a Runge domain $\Omega$ of an
affine algebraic variety $S$ into a projective algebraic manifold $X$ is a
uniform limit of Nash algebraic maps $f_\nu$ defined over an exhausting
sequence of relatively compact open sets $\Omega_\nu$ in $\Omega$. A relative
version is also given: If there is an algebraic subvariety $A$ (not necessarily
reduced) in $S$ such that the restriction of $f$ to $A\cap\Omega$ is algebraic,
then $f_\nu$ can be taken to coincide with $f$ on $A\cap\Omega_\nu$. The main
application of these results, when $\Omega$ is the unit disk, is to show that
the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric of a
quasi-projective algebraic manifold $Z$ are computable solely in terms of the
closed algebraic curves in $Z$. Similarly, the $p$-dimensional Eisenman metric
of a quasi-projective algebraic manifold can be computed in terms of the
Eisenman volumes of its $p$-dimensional algebraic subvarieties. Another
question addressed in the paper is whether the approximations $f_\nu$ can be
taken to have their images contained in affine Zariski open subsets of $X$. By
using complex analytic methods (pluricomplex potential theory and H\"ormander's
$L^2$ estimates), we show that this is the case if $f$ is an embedding (with
$\dim S<\dim X$) and if there is an ample line bundle $L$ on $X$ such that
