Ax, Kochen, and Ershov found the complete axiomatization of the theory of the field Qp of p-adic numbers, for each prime p. A proof of the completeness of their axiomatization usually goes through two stages: first an analysis of the model theory of henselian valued fields of equal characteristic zero, then a smidgen of structure theory of CDVRs in mixed characteristic. The same pattern applies more generally to axiomatizations of the theories of finitely ramified valued fields. The role of the CDVRs at the core of such theories is played in the infinitely ramified case by a different kind of valued field: maximal and R-valued. Moreover, the theory of these core (valued) fields is an invariant of the theory of the original valued field. When their residue fields are perfect, these are tame and some known model theory may be applied, although there are still plenty of open questions, including a "composition AKE" problem. In the imperfect case, things are yet wilder. This is ongoing joint work with Jahnke and Ketelsen.