The fundamental intuition in the model theory of valued fields is that a valued field with appropriate closure properties is controlled in some sense by its value group and residue field. The classical theorem of Ax-Kochen and Ersov states a version of this at the level of the theory of a henselian valued field of characteristic (0,0). In this talk, I will discuss some theorems for algebraically closed valued fields and real closed valued fields which give conditions on the value group and residue field for the isomorphism type of the field itself to be fixed. I will also explain how these results also apply to more general henselian valued fields. I will review the background on valued fields, so this abstract should make sense by the end of the talk, if not before.