There are three primary schools of thought on the Deuteronomistic History. The first to postulate the DH was Martin Noth, who went against the grain of previous scholars of the Old Testament, and argued that rather than the books of Joshuah-2 Kings being the work of various authors and/or redactors that there was actually only a single author/redactor whom he called the Deuteronomistic Historian (Dtr) and whose work he labelled the Deuteronomistic History (DH). Instead of seeing many strands of tradition and compositions, Noth recognized a unification of these works which in his estimation represented five different “histories” of Israel with singular authorial intent. This singular tradent compiled numerous sources (including citing some by name) and composed his work as the theological history of Israel from the end of Moses' life to the end of the monarchy. According to Noth, it was written shortly after the release of Jehoiachin from imprisonment at the hands of the Babylonians and was intended to help Israel reflect upon the reason for their exile and God's just judgment.
Following the work of Noth, several scholars (von Rad and Wolff) noted what they believed to be redactional activity accomplished after the proposed date of the Dtr of Noth's theory. There were also issues with the largely negative assessment of Noth concerning the authorial intent of his Dtr.
This in turn led to two further schools of thought: the so-called “Harvard school” and the “Göttingen school.” The former was led by the work of Frank Moore Cross who postulated a double redaction of the DH. Essentially Cross held to Noth's theory of the more negative view of the Dtr, but added a second view for this author/editor: “grace” (DOT:HB 223). He also believed there was a later author/editor whom he labelled Dtr2 in contradistinction to Dtr1. The work of Dtr1 was (according to this school) composed sometime around the reign of Josiah and he held to hope for redemption because of the Josainic reforms. While Dtr1 held to the double message of judgment/grace (with the emphasis on the latter as the hope of Israel); Dtr2 was believed to have written during the exile and appended (and inserted into the DH of Dtr1) passages which indicated the inevitability of exile despite the earlier Josianic reforms. This was an attempt to explain the notions of judgment, hope and finally judgment.
A German scholar, R. Smend Jr., founded the “Göttingen school” of thought on the DH distinct from the “Harvard school” of Cross. Smend and his “school” postulated that Noth's Dtr was an exilic initial and primary compiler whom he called DtrG (or DtrH). This work was added to by a later redactor (whome he called DtrN) who had a particular nomistic intent to his writing and thus emphasized the law and problems of foreign presence and influence in Israel. One of Smend's students felt that Smend's theories did not sufficiently deal with all of the material of the DH and so he added a further (and later) redactor whom he labelled DtrP as the prophetic Deuteronomist. This final redactor made much of the reign of Manasseh according to Dietrich. However, it remains questionable (even among those of the “Göttingen school”) whether there really is any distinction to be made between DtrH and DtrP.
Richter, Sandra L. “Deuteronomistic History,” Dictionary of the Old Testament: Historical Books (Eds. Bill T. Arnold and H. G. M. Williamson. Downers Grove, IL: InterVarsity Press, 2005): 219-230.