  We prove a suite of results classifying holomorphic maps between
configuration spaces of Riemann surfaces; we consider both the ordered and
unordered setting as well as the cases of genus zero, one, and at least two. We
give a complete classification of all holomorphic maps
$\operatorname{Conf}_n(\mathbb{C})\to \operatorname{Conf}_m(\mathbb{C})$
provided that $n\ge 5$ and $m\le 2n$ extending the Tameness Theorem of Lin,
which is the case $m = n$. We also give a complete classification of
holomorphic maps between ordered configuration spaces of Riemann surfaces of
genus at most one (answering a question of Farb), and show that the higher
genus setting is closely linked to the still-mysterious ``effective de Franchis
problem''. The main technical theme of the paper is that holomorphicity allows
one to promote group-theoretic rigidity results to the space level.
