The belief in Mosaic authorship of the Torah is found in the Torah itself which states, "So Moses wrote down this law and gave it to the priests, the sons of Levi, who carried the ark of the covenant of the LORD, and to all the elders of Israel." Deuteronomy 31:9:KJV
The belief of Mosaic authorship excluding the Bible's testimony is first found explicitly expressed in the Talmud, a collection of Jewish traditions and exegesis said supposedly to date to the time of Moses himself. The commentary section of the talmud though dates to 500 AD, and indicate that the authors of the later books of the Hebrew bible already accepted the idea that Moses had written the Torah. The Talmudic commentators advanced several versions of just how Moses came to write the Torah, ranging from direct dictation by God to a less direct divine inspiration stretching over the forty years in the wilderness.
Origins and nature of the tradition
The Torah itself makes a statement of authorship in various verses. Notable among these is Deuteronomy 31:9 ,Deuteronomy 31:24-26 , describing how Moses writes "this law" on a scroll and lays it beside the ark of the Covenant. Similar passages include, for example, Exodus 17:14, "And Jehovah 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 Jehovah, and rose up early in the morning, and builded an altar under the mount, and twelve pillars, according to the twelve tribes of Israel;" and Exodus 34:27, "And Jehovah said unto Moses, Write thou these words: for after the tenor of these words I have made a covenant with thee and with Israel." Also Leviticus 26:46 states, "These are the decrees, the laws and the regulations that the LORD established on Mount Sinai between himself and the Israelites through Moses"
Joshua, Kings, Chronicles, Ezra and Nehemiah all contain verses implying belief in Mosaic authorship of the Torah, indicating that the belief possibly existed during the time of Joshua, his book is said to have been written around the 1300s bc. Also because of the belief in Mosaic authorship is found in the book of Nehemiah it can be certain that this belief existed during the Post-Exilic period. It was certainly well established by the time of the Talmud (c. 200-500 AD), the authors of which held that Moses received the Torah during the forty years of wandering in the wilderness. The early Christian church with its Jewish roots accepted the Torah, and Mosaic authorship, as part of its own spiritual inheritance.
The Authors of the Gospels also believed in Mosaic authorship, as did the famous Roman historians Philo, and Josephus Flavius.
The Mosaic tradition in the modern age
Many of the supposed contradictions and inconsistencies said to exist in the Torah as noted by critical scholars have been well noted by the classical Jewish sources (and in part form the basis of the Oral Torah). R' David Zvi Hoffman in his commentary to Leviticus made use of rabbinic homiletical and exegetical interpretations as well as some of his own insights to explain the text in light of the difficulties noted by the critics. He also authored a book Die wichtigsten Instanzen gegen die Graf-Wellhausensche Hypothese (2 vols., 1903/1916 translated into Hebrew and available [here| http://www.daat.ac.il/daat/tanach/reayot/tohen-2.htm]) 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.
Another scholar Benno Jacob developed a theory concerning the internal rhythm of the Bible, which is expressed by the repetition of key words in set numbers in the narratives of the Torah and its laws. The programmatic statement in his 1916 book, Quellenscheiden und Exegese im Pentateuch, illustrates his concerns:
The Bible’s means of representation (Darstellungsweise) may be termed the semi-poetic or dichotomistic. It proceeds like poetry, but without its strict measure [i.e., meter], employing instead paired thoughts, patterns of words and clauses and syntax, in doublets, parallels and contrasts; it is rooted, when all is said and done, in the Semitic [way of thought], which grasps matters dichotomously. This manner of seeing, conceiving and representing dominates the Hebrew language and literature in its entirety, to its subtlest manifestations.
Dr. Mordechai Breuer's approach is as follows. The Torah must speak in "the language of men." But the wisdom that God would bestow upon us cannot be disclosed in a straightforward manner. The Torah therefore resorts to a technique of multivocal communication. Each strand in the text, standing on its own, reveals one aspect of the truth, and each aspect of the truth appears to contradict the other accounts. An insensitive reader, noticing the tension between the versions, imagines himself assaulted by a cacophony of conflicting voices. The perceptive student, however, experiences the magnificent counterpoint in all its power. To use Rabbi Breuer's example: Genesis 1 (the so-called P account) describes one aspect of the biblical understanding of creation; Genesis 2 (the so-called J version) presents a complementary way of apprehending God's creation of the world and of man. Each text, isolated from the other, would offer a partial, hence misleading, doctrine of creation. In their juxtaposition, the two texts point the reader toward an understanding of the whole.
Egyptian terminology in the Torah
Certain forms in standard Biblical Hebrew are borrowed from second-millennium Egyptian. This is seen in how the birth naratives of Moses are full of words of Egyptian origin (instead of Hebrew origin), examples include, basket, bulrushes, pitch, reeds, river, and river-bank. Also Genesis 41 which speaks of the Pharaoh's dream during the time of Joseph uses several Egyptian words such as, the word for magician. The Torah is also known to use phrases which are of Egyptian origin in which the words are translated word for word One may infer that these forms were adopted during the sojourn and were made a permanent part of standard Hebrew by their inclusion in the Pentateuch. 
Antiquity of the Hebrew found in the Torah
The Torah has the tendency to use some archaic Hebrew forms, which suggests that its origin antedates the Israelite monarchies. For example, Genesis has a common 3rd person singular pronoun form -hw; Joshua and later works breaks this into masculine and feminine forms. Also of note, is that the word "Goshen", which is mentioned extensively throughout the Torah, is only used in the pre-monarchy texts (the latest reference is found in Joshua 15); all subsequent biblical references to the area do not refer to this. Some of the poetic material preserved in the Torah is incredibly ancient, and reflects syntax and semantic usages that disappeared later in the Old Testament historical period. The poetry of the Bible, like that of other Northwest Semitic literaruters, employs a language which differs in various ways from the language of prose, reflecting, in general, an earlier stage of Hebrew and with a closer affinity in language, style, and content with neighboring dialects, especially those to the north." Notable among the biblical passages that best reflect Archaic Hebrew are the Blessings of Jacob (Gn 49), the Song of Moses (Ex 15), Balaam (Nm 23-24), the Oracles of and the Poem of Moses (Dt 32), the and of Moses (Dt 33). One also finds widespread use of the third person pronominal suffix -mo (e.g. Ex 15.5,7), the second person feminine suffix -ky, the third person singular masculine suffix -h instead of -w (e.g. Gen 49.11), infinitive absolute with temporal value (e.g. Ex 15.6), zo and zu used as relative particles (Ex 15.13), use of the negative bal instead of lo, the verbal suffix -t in the third person feminine (e.g. Dt 32.36) traces of the old case endings in nouns suffixed by -i or -o in the construct state (e.g. Gen 49.11; Nm 23.18). "Expressions used almost exclusively in poetry include hapax legomena and other rare words, which tend to be concentrated in the oldest biblical texts. Generally it may be said that these items existed during the archaic period of the language, later disappearing from normal use...The occurrence of so many lexical items of this kind in a single passage is evidence of its antiquity." 
In 1979, two silver scrolls that were used as amulets, inscribed with portions of the well-known Priestly Blessing of the Book of Numbers were discovered in a burial cave near Jerusalem. These scrolls have been dated to close to 600 BC based on late Iron Age artifacts found in the undisturbed area of the tomb where they were located.
Should these datings be correct the date of the Torah would be much older than what most Biblical critics think it is, it would also mean it would be more likely for the Torah to have been written by Moses because of this earlier date.
Also based on paleographic evidence Erik Waaler, in his book "A revised date for Pentateuchal texts?" published in 2002, dates the amulets somewhat earlier than the other artifacts in the cave (725-650 BC).
The work of the American school of biblical archaeologists such as William F. Albright and Cyrus Gordon have confirmed that Genesis and Exodus are firmly grounded in the material reality of the second millennium. The Torah accurately portrays second-millennium legal and social customs. An example of this how the price of 20 shekels price for Joseph was the going price for a slave during the first half of the 2nd millennium, whereas in the 2nd half of that millennium the price had gone up to 30 shekels. The legal and cultural patterns present in the Patriarchal narratives simply no longer existed in exilic or post-exilic times.Furthermore, the customs manifested by Abraham & the patriarchs that descend from him are most closely matched by the society illustrated in the Nuzi tablets (of a Hurrian background peoples), dated in the first half of the 2nd millennium BC 
Furthermore, for those who doubt the historicity of the Exouds completely, by suggesting that it was created only in the sixth to fifth century B.C. post-exilic era, a question must be asked regarding Ramesses and Pithom, the cities on which the Hebrews labored, according to Exodus. Why did the biblical editors or redactors refer specifically to Ramesses, when in their own era and for some three centuries earlier the capital of Egypt had been Tanis, a city well known and often referred to in the Old Testament? From the Book of Judges onwards, Tanis is consistently referred to as Egypt's capital. Why would a biblical editor insert Ramesses into a newly composed story when that city no longer existed in Egypt and had not been Pharaoh's residence or the capital for the previous four or five centuries? ...Tanis had been the Egyptian capital throughout nearly the entire span of Israel's monarchic period. What sense would it make for Jews familiar with Saite Egypt to invent a story about an oppressive pharaoh who had compelled their ancestors to labor on his cities, and why fix on Ramesses for this role? In Dynasty XXVI Pharaoh's capital was Sais, and even more pointedly, Jewish exiles in Egypt were valued for their mercenary skills and not consigned to compulsory brick making.
There is a trend among scholars to view the Pentateuch as a literary unit again. Scholars are admitting that the way the books use common words, phrases and motifs, parallel narrative structure, and deliberate theological arrangement of literary units for teaching and memorization support viewing the five books as a literary whole.
- Joshua 1:7-8
- 1 Kings 2-3 and 2 Kings 23:21 and 25
- 2 Chronicles 8:13, 34:14 and 35:12
- Ezra 3:2 and 6:18
- Nehemiah 8:1 and 13:1
- Livingston, Herbert G. (1974). The Pentateuch in Its Cultural Environment. Baker Book House. ISBN 0801055407.
- "Emunah U-Madda Be-Parashanut Ha-Mikra," Deot, Cheker Ha-Mikra Be-Machshavah Ha-Yehudit Ha-Datit He-Chadashah, 11 (1959):18-25, 12 (I960): 13-27.
- Israel in Egypt: The Evidence for the Authenticity of the Exodus Tradition, James K. Hoffmeier, Oxford: 1997. pages 139-140, 151
- Israel in Egypt: The Evidence for the Authenticity of the Exodus Tradition, James K. Hoffmeier, Oxford: 1997. page 88
- Israel in Egypt: The Evidence for the Authenticity of the Exodus Tradition, James K. Hoffmeier, Oxford: 1997. pages 151
- Garrett, Duane (1991). ReThinking Genesis. Baker Pub Group, 84 to 85. ISBN 0801038375.
- The Redaction of Genesis, Gary A. Rendsburg, Eisenbraus:1986 page 114
- Exodus: The Egyptian Evidence, E.S. Frerichs and L.H. Lesko (eds), Eisenbrauns:1997. page 44
- Israel in Egypt: The Evidence for the Authenticity of the Exodus Tradition, James K. Hoffmeier, Oxford: 1997. page 21
- A History of the Hebrew Language, Angel Saenz-Badillos (trans. by John Elwolde), Cambridge:1993. pages 56-61
- "Who Wrote the Bible?". The Naked Archaeologist.
- "Archaeology and the Patriarchs"
- Israel in Egypt: The Evidence for the Authenticity of the Exodus Tradition, James K. Hoffmeier, Oxford: 1997. page 84
- The Bible and the Ancient Near East, Cyrus Gordon and Gary Rendsburg, Norton:1997 (4th ed). pages 109-130
- Andrew Hill & John H. Walton, A Survey Of The Old Testament, (Grand Rapids, MI: Zondervan Publishing House, 1991), 81.