We study the class of sets definable by first-order formulae in henselian valued fields. The guiding principle, building on classical work by Ax-Kochen and Ershov, is that definable sets in well-behaved henselian valued fields are governed by those of the residue field and the value group. We present a new theorem generalizing these classical ideas. We then discuss when a valuation is so intrinsic to the field that its valuation ring is definable using just the arithmetic of the field. This is closely linked to a conjecture by Shelah that considers fields for which the class of definable sets has restricted combinatorial complexity, i.e., no formula has the independence property. The conjecture predicts that any infinite such field is separably closed, real closed, or admits a nontrivial henselian valuation. We present the state of the art regarding the conjecture, including a theorem that any NIP henselian valued field satisfies an Ax-Kochen/Ershov principle. Arithmetic definability of valuations turns out to be a key tool in this context.