  Let K be an algebraically closed, complete non-Archimedean field. The purpose
of this paper is to carefully study the extent to which finite morphisms of
algebraic K-curves are controlled by certain combinatorial objects, called
skeleta. A skeleton is a metric graph embedded in the Berkovich analytification
of X. A skeleton has the natural structure of a metrized complex of curves. We
prove that a finite morphism of K-curves gives rise to a finite harmonic
morphism of a suitable choice of skeleta. We use this to give analytic proofs
of stronger "skeletonized" versions of some foundational results
ofLiu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of
curves. We then consider the inverse problem of lifting finite harmonic
morphisms of metrized complexes to morphisms of curves over K. We prove that
every tamely ramified finite harmonic morphism of \Lambda-metrized complexes of
k-curves lifts to a finite morphism of K-curves. If in addition the
ramification points are marked, we obtain a complete classification of all such
lifts along with their automorphisms. This generalizes and provides new
analytic proofs of earlier results of Sa\"idi and Wewers. As an application, we
discuss the relationship between harmonic morphisms of metric graphs and
induced maps between component groups of N\'eron models, providing a negative
answer to a question of Ribet motivated by number theory.
  This article is the first in a series of two. The second article contains
several applications of our lifting results to questions about lifting
morphisms of tropical curves.
