  We present a framework for tame geometry on Henselian valued fields which we
call Hensel minimality. In the spirit of o-minimality, which is key to real
geometry and several diophantine applications, we develop geometric results and
applications for Hensel minimal structures that were previously known only
under stronger, less axiomatic assumptions. We show existence of
t-stratifications in Hensel minimal structures and Taylor approximation results
which are key to non-archimedean versions of Pila-Wilkie point counting,
Yomdin's parameterization results and to motivic integration. In this first
paper we work in equi-characteristic zero; in the sequel paper, we develop the
mixed characteristic case and a diophantine application.
