Mosaic authorship is the traditional belief that the five books of the Torah or Pentateuch were authored by Moses sometime between 13th and 17th century BCE. Mosaic authorship was accepted almost without question by both Jews and Christians until the 17th century AD, but the rise of secular scholarship eventually led to its rejection by many. It still has followers among conservative religious scholars, who seek to reconcile it with modern scholarly findings.
Deuteronomy 31:9 and Deuteronomy 31:24-26 describe how Moses writes "torah" (instruction) on a scroll and lays it beside the ark of the Covenant. Similar passages include, for example, Exodus 17:14, "And YHWH said unto Moses, Write this for a memorial in a book, and rehearse it in the ears of Joshua, that I will utterly blot out the remembrance of Amalek from under heaven;" Exodus 24:4, "And Moses wrote all the words of YHWH, and rose up early in the morning, and builded an altar under the mount, and twelve pillars, according to the twelve tribes of Israel;" Exodus 34:27, "And Yahwh said unto Moses, Write thou these words, for after the tenor of these words I have made a covenant with thee and with Israel;" and Leviticus 26:46 "These are the decrees, the laws and the regulations that the LORD established on Mount Sinai between himself and the Israelites through Moses."
The first unequivocal statement Mosaic authorship is contained in the Talmud, c. 200-500 CE, where the rabbis discussed exactly how the Torah was transmitted to him. In the Babylonian Talmud Gittin 60a it is written "Said R' Yochanan, the Torah was given in a series of small scrolls," implying that the Torah was written gradually and compiled from a variety of documents over time. This may be an attempt to account for the composite appearance of the five books, but a contrary Talmudic opinion holds that the entire Torah was given at one time.
The rise of modern scholarship and the abandonment of the Mosaic hypothesis
The origin of the Torah was not a matter of interest to traditional scholars, and its authorship by Moses was accepted without serious question. This changed with the European Enlightenment, when the rise of secular scholarship produced an anti-traditional fervor and a willingness to subject even the Bible to the test of empirical evidence. By the 19th century the traditional view was no longer entertained by mainstream scholars, and in the closing decades of the 19th century Julius Wellhausen put forward the Documentary hypothesis, the theory that the Pentateuch had its origins in four source documents composed at various times during the 1st millennium BCE and combined into the final Torah ca. 450 BCE.
Wellhausen's formulation maintained a near-consensus for most of the 20th century, but by the mid-20th century multiple challenges began to appear. Today Pentateuchal studies as a scholarly field is in ferment, with competing versions from all three possible models, the documenary (the Torah as a compilation of originally separate but complete books), the supplementary (a single original book, supplemented with later additions/deletions), and fragmentary (many fragmentary works and editions); but there remains a consensus that it is the work of many hands and many centuries, and that its final form belongs to the middle of the 1st millennium BCE.
Modern Jewish scholarship within the Mosaic tradition
Contradictions and inconsistencies in the Torah were noted by classical Jewish sources, and in part form the basis of the Oral Torah. In modern times Rabbi David Zvi Hoffman, in his commentary to Leviticus, made use of rabbinic homiletic and exegetical interpretations as well as some of his own insights to explain the difficulties noted by Wellhausen and othe critics and defend Mosaic authorship. His Die wichtigsten Instanzen gegen die Graf-Wellhausensche Hypothese (2 vols., 1903/1916) pointing out several difficulties in the Wellhasuen hypothesis, most notably in his theory that the Priestly code, and hence the Jewish conception of monotheism, was of late post-exilic redaction. While his approach to biblical investigation was essentially the result of the conditions of his time and place, they have stood the test of time and are still studied.
Several attempts have been made to reconcile the results of modern scholarship with the traditional belief that Moses wrote the Torah. For instance, Dr. Mordechai Breuer believes that "the Torah must speak in the language of men." Therefore Breuer postulates that the Torah resorts to a technique of multi-vocal communication: the sum of texts that appear dissimilar in fact offers a powerful counterpoint.  Similarly, the contemporary Jewish scholar Menachem Mendel Kasher points to certain traditions of the Oral Torah which show Moses quoting Genesis prior to the epiphany at Sinai; based on a number of Bible verses and rabbinic statements, he suggests that Moses had certain documents authored by the Patriarchs that he used made use of when redacting that book.
A major contemporary intellectual contribution in this vein is Revelation Restored, by Dr. David Weiss Halivni. Halivni develops a theory of Chate'u Yisroel (literally, "Israel has sinned"): "According to the biblical account itself, the people of Israel forsook the Torah, in the dramatic episode of the golden calf, only forty days after the revelation at Sinai. From that point on, until the time of Ezra, the scriptures reveal that the people of Israel were steeped in idolatry and negligent of the Mosaic law. Chate’u Yisrael, as a theological account, explains that in the period of neglect and syncretism the Torah of Moses became blemished and maculated." According to Halivni, this process continued until the time of Ezra (c.450 BC), when finally, upon their return from Babylon, the people accepted the Torah. It was at that time that the previously rejected, and therefore maculated, text of the Torah was recompiled and edited, by Ezra and his “entourage.” That this is what happened is attested to in the books of Ezra and Nehemiah. Halivni supports his theory with Talmudic and Midrashic sources which indicate that Ezra played a certain role in editing the Torah. He further states that while the text of the Pentateuch was corrupted, an oral tradition preserved intact many of the laws of the Bible. This is why the Oral law appears to contradict the Biblical text in certain details.