It is increasingly common to hear and read about the "Hebrew Bible" and the "theology of the Hebrew Scriptures."
What distinguishes this from Old Testament theology?
Old Testament theology is a subset of the discipline of biblical theology. The term "Old Testament" reflects the Christian origins and heritage of the discpline (see Jon D. Levenson, “Why Jews Are Not Interested in Biblical Theology,” in The Hebrew Bible, the Old Testament, and Historical Criticism: Jews and Christians in Biblical Studies [Louisville: Westminster John Knox Press, 1993]: 33-61).
The alternative language reflects the changing shape of the field. Some Christian scholars treat the "Old Testament" as a discrete theological witness (not as the first of a two-part theology). Additionally, there are Christian scholars in dialogue with Jewish scholars discussing their shared Scriptural traditions. In these two situations, scholars use "Hebrew Bible" to good effect. Furthermore, Jewish scholarship in the discpline uses this or similar language (e.g., Jewish Bible theology, a theology of the Tanak, etc.).
What language should I use in my own work?
Context determines the more appropriate language to use when writing in this area. Students at the Southern Baptist Theological Seminary are most frequently engaging the discipline of Old Testament theology from within the Christian tradition and treating it as constituting the first of a two-part theology. For such activity, students are encouraged to take up the language of Old Testament theology.