For a language $\mathcal{L}$, define the Herbrand universe to be the set of closed terms (alternatively ground terms) of $L$.

A structure $\mathfrak{M}$ for $\mathcal{L}$ is a Herbrand structure if the domain of $\mathfrak{M}$ is the Herbrand universe of $\mathcal{L}$. This fixes the domain of $\mathfrak{M}$, and so each Herbrand structure can be identified with its interpretation, leading to the alternative nomenclature of Herbrand interpretation.

A Herbrand model of a theory $T$ is a Herbrand structure which is a model of $T$.