I will discuss valued differential fields. What parts of valuation theory go through for these objects, under what conditions? Is there a good differential analogue of Hensel’s Lemma? Is there a reasonable notion of differential-henselization? What about differential-henselianity for systems of algebraic differential equations in several unknowns?

I will mention results as well as open questions. The results are part of joint work with Matthias Aschenbrenner and Joris van der Hoeven, and have turned out to be useful in the model theory of the valued differential field of transseries.