This chapter considers H-fields, pre-differential-valued fields with a field ordering that interacts with the valuation and derivation. Axiomatizing this interaction yields the notion of a pre-H-field; H-fields are d-valued pre-H-fields. The chapter begins by upgrading some basic facts on asymptotic fields to pre-d-valued fields; for example, algebraic extensions of pre-d-valued fields are pre-d-valued, not just asymptotic. It then adjoins integrals to pre-d-valued fields of H-type. It shows that every pre-d-valued field of H-type has a canonical differential-valued extension. It also adjoins exponential integrals to pre-d-valued fields of H-type. Finally, it describes Liouville closed H-fields, and especially the uniqueness properties of Liouville closure.
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