I didn't answer this question for a long while because I assumed that since there were so many answers already provided that at least one of them must have answered it truthfully with the best evidence available. Unfortunately that was not the case, so I'm answering this question despite it having already accepted a different answer.
The truth is that every bit of evidence we have says that the Torah underwent changes. Most of these changes were not significant, but occasionally a small change makes a significant difference. [Of the Biblical textual differences, most of them are names of places, spelling differences, ages or heights of people.] For example, by the time we reach the flood during the time of Noah, the "Official" Torah has existed for 1,656 years, whereas other ancient versions has the world having existed for 2,242 years. Any differences that were theological in nature were very rare.1
But what is also true that after a certain time in history, the Torah/Biblical text stopped changing, scribal practices started conforming to each other more, and every Jewish community ended up with the same text (minus incidental spelling differences of the same words). By the time of Muhammad, every [known] Jewish community had the same Biblical text [as in any differences between texts didn't change anything theologically].
But the Dead Sea Scrolls predate Muhammad, and they predate any other known Biblical texts. These Dead Sea Scrolls have shown convincing evidence of corruptions in our received text which I will show below.
The version of the text that Jews have been relying upon for the past 1,000 years is called the Masoretic text. This is the "official" version of the Biblical text for religious Jews. Unfortunately, this "text" in manuscript form is only 1,000 years old. The oldest [surviving and unfortunately not complete] manuscript we have of the Masoretic text is the Aleppo Codex, which has been digitized and can be viewed/read here: http://aleppocodex.org/ Note that there are other types of Masoretic texts, but the Aleppo Codex was used by the Rambam, and he claimed that all [scholars] relied upon it, and this codex was treasured and highly valued by all of world Jewry.
However, we know that the Torah is more than 1,000 years old, so what about more ancient manuscripts? Well, we didn't have any older Hebrew manuscripts [before the discovery of the Dead Sea Scrolls], but we had other manuscripts of ancient Greek, Aramaic, Samaritan, and Latin translations of the Hebrew Bible. The problem? There were lots of differences between these translations and the Masoretic Text. For example, the Greek translation, called the Septuagint, had extra verses in the Torah, had extra books (such as Ben Sirach, Ecclesiasticus, Maccabees, etc), and had extra chapters for books like Daniel. Rabbis in historical discourses between them and Catholics said those changes were caused by the process of the translation, or due to books being included for political reasons etc. But the problem is that most times, all these different ancient translations would agree with each other, and it was only our Masoretic text that would be different.
An example would be Genesis 4:8. The Masoretic text says the following:
ח וַיֹּאמֶר קַיִן, אֶל-הֶבֶל אָחִיו; וַיְהִי בִּהְיוֹתָם בַּשָּׂדֶה,
וַיָּקָם קַיִן אֶל-הֶבֶל אָחִיו וַיַּהַרְגֵהוּ. 8
And Cain spoke unto Abel his brother. And it came to pass, when they were in the field, that Cain rose up against Abel his brother, and slew him.
Notice something missing? The Masoretic text states "And Cain said to Abel....and they were in the field." The Masoretic text doesn't state what Cain says. But the Greek Septuagint, the Samaritan Torah, the Aramaic Peshitta, and every other ancient manuscripts have these extra words: "And Cain said to Able let us go to the field..." So we have instances of every older translation of the Hebrew text being in agreement, and the Masoretic text being the only one that's different.
And then the Dead Sea Scrolls were discovered. And while they are also missing the words of Cain, there are literally thousands of differences between the Dead Sea Scrolls and the Masoretic text. And many times, the Dead Sea Scrolls agrees with the older translations we've been talking about and disagrees with the Masoretic text we have received. A good example is this question: How tall was Goliath?
1st Samuel 17:4
Masoretic Text:
ד וַיֵּצֵא אִישׁ-הַבֵּנַיִם מִמַּחֲנוֹת פְּלִשְׁתִּים, גָּלְיָת
שְׁמוֹ מִגַּת: גָּבְהוֹ, שֵׁשׁ אַמּוֹת וָזָרֶת. 4
And there went out a champion from the camp of the Philistines, named
Goliath, of Gath, whose height was six cubits and a span.
So according to the Masoretic text, Goliath was roughly 9 feet tall, which is quite miraculous. But The Septuagint, Josephus, other ancient texts and the newly discovered Dead Sea Scrolls read the following:
And there went out a champion from the camp of the Philistines, named
Goliath, of Gath, whose height was four cubits and a span.
Goliath has shrunk from 9 feet tall to 6'9; which is definitely impressive...but not miraculous. So with the Dead Sea Scrolls we have strong corroborating evidence that our received Masoretic Text is corrupt with how tall Goliath was. And yet Jews disregard this very strong evidence and continue to teach their kids that Goliath was a 9 foot tall giant, when the reality is that he was probably about 6'9.
Another famous example of where ancient translations and the Dead Sea Scrolls agree against the Masoretic text is Psalm 145, known as Ashrei, which is recited every Shabbat. The Psalm is an Acrostic, with every line beginning with a different letter of the Hebrew alphabet, except for the letter Nun. The Babylonian Talmud discusses this:
Berakhot 4b:
But [Psalm Psalm 145 is selected] because it contains both features.
R. Johanan said: Why is there no verse beginning with the letter Nun
in that Psalm ? Because it would refer to the downfall of Israel's
enemies; as it is written, "Fallen (Naphelah) is the virgin of Israel,
she shall no more rise" (Amos v. 2).
But the Septuagint, Peshitta, Vulgate, and Dead Sea Scrolls are in agreement that there should be a line with a Nun, and here it is:
"Faithful is the LORD in all His ways, and merciful in all His works" (Hebrew: "נאמן אלוהים בדבריו וחסיד בכל מעשיו").
But what about examples where all the older manuscripts agree, but we didn't find a copy of it in the Dead Sea Scrolls? Well, we have an example that appears to be a clear case of textual corruption:
Masoretic Text: 1st Samuel 13:1
א בֶּן-שָׁנָה, שָׁאוּל בְּמָלְכוֹ; וּשְׁתֵּי שָׁנִים, מָלַךְ
עַל-יִשְׂרָאֵל. 1
Saul was one years old when he began to reign; and two years he
reigned over Israel.
We all know that Saul had to be more than a year old when he became king, and we know that he reigned for more than 2 years. Yet Jewish tradition explains away these discrepancies rather than try to discuss possible corruptions. Take a look at Rashi's attempts to gloss over the textual corruption on that verse:
Saul was a year in his reign: (lit., a year old.) Our Rabbis of sainted memory said: Like a one year old, who did not experience the taste of sin (Yoma 22b). It may also be interpreted thus: Saul was a year in his reign, i.e., in the first year in which he was made king (and he reigned two years over Israel), and in the first year...
And yet the Septuagint, Vulgate, and every other ancient text has this text:
Saul was thirty years old when he began to reign; and forty two years he
reigned over Israel.
Unfortunately we didn't find a copy of the Dead Sea Scrolls that contains this verse, but considering our other examples thus far, it would more than likely match this text.
All of these examples thus far have shown differences of little theological importance, so let's show one that has big implications.
Deuteronomy 32:8
Masoretic Version:
ח בְּהַנְחֵל עֶלְיוֹן גּוֹיִם, {ס} בְּהַפְרִידוֹ בְּנֵי אָדָם; {ר}
יַצֵּב גְּבֻלֹת עַמִּים, {ס} לְמִסְפַּר בְּנֵי יִשְׂרָאֵל. {ר} 8
When the Most High gave to the nations their inheritance, when He
separated the children of men, He set the borders of the peoples
according to the number of the children of Israel.
Dead Sea Scrolls:
ח בְּהַנְחֵל עֶלְיוֹן גּוֹיִם, {ס} בְּהַפְרִידוֹ בְּנֵי אָדָם; {ר}
יַצֵּב גְּבֻלֹת עַמִּים, {ס} לְמִסְפַּר בְּנֵי אֱלֹהִים. {ר} 8
When the Most High gave to the nations their inheritance, when He
separated the children of men, He set the borders of the peoples
according to the number of the "children of Elohim"
So while the Dead Sea Scrolls prove that there was a lot of differences in the Biblical text, we stop seeing these kinds of large changes after the time of the Dead Sea Scrolls. The Dead Sea Scrolls are dated to be written around 100-300BCE, but after that, the texts seem to be much more fixed and stop having any relevant differences. Maybe the scribes got more oversight, or were better at their craft. Maybe differing manuscripts were destroyed or lost to history with the destruction of the Temple and therefore could no longer be copied or spread around. Either way, there is credibility to the claim that our text has not been a perfect transmission. But most of the "changes" wouldn't matter religiously, to Muhammad, or to Jews.
