Liouville closed H-fields

H-fields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending image and fields of transseries over image are H-fields.) We prove basic facts about the location of zeros of differential polynomials in Liouville closed H-fields, and study various constructions in the category of H-fields: closure under powers, constant field extension, completion, and building H-fields with prescribed constant field and H-couple. We indicate difficulties in obtaining a good model theory of H-fields, including an undecidability result. We finish with open questions that motivate our work.