The main feature of the upcoming holiday of Simchat Torah is the conclusion of the annual reading of Torah and the symbolic beginning of the new cycle (“symbolic” because although some verses from Genesis are read, the actual parshah/section is read in its entirety on the Shabbat after Simchat Torah is over).
Judaism introduced to the world the concept of the public reading of Scripture. Previously, the text and learning of scripture was always held within the circle of priests or other clergy. This was even true of Judaism in the “First Temple” or “Old Testament” period (although “OT” isn’t a term usually used by we Jews ourselves) — the only ones learned enough in Torah to teach it were the kohanim. There must have been some public learning, though, because of quotations and references to Torah that appear in the “Prophets” and “Writings.” The development of the format of public reading of Torah was gradual, occurring over many centuries, until we arrived at the system in place today, of sequential weekly readings that are repeated annually. Public reading later passed into Christian and Muslim tradition.
One of the differences between the “orthodox” and Protestant churches has to do with free access by any congregant of the latter to reading the Bible and interpreting it more or less in your own way (within certain guidelines, of course). Protestant traditions encourage individual, personal reading and understanding of Scripture, combined with instruction from clergy and trained teachers. A Methodist friend, for example, told me of her “quiet time” each morning — 10 minutes of reading Scripture and talking to G-d about the coming day. In Catholicism, on the other hand, there’s public reading of the Bible and a congregant might own one (now, with the invention of the printing press), but the interpretation of Scripture is strictly in ecclesiastical hands.
As we know, the public reading of the Torah is called in Hebrew the “Kriyah,” or “Kriyat ha-Torah.” “Kriyah” means recitation (i.e. reading aloud, as distinguished from the private study of it, which is called “talmud Torah” — “learning Torah”).
In Arabic, “kriyah” is — “Kur’an” (or: Qur’an). Yes, the name of the Muslim scripture means “The Reading” — i.e. it’s that which is read aloud in public services. I don’t know at this time whether there’s a specific annual sequence of readings, as in Judaism, but just as the reading of Torah is divided into sections called “sidrahs,” each of which is titled with a word that usually appears in the first sentence (with some exceptions), the Kur’an is divided into sections called “surrahs,” each of which is titled…(for the completion of this sentence, read the preceding one)! Some scholars  have opined that even the word “surrah” was originally “sidrah,” although there’s no empiric proof of this (yet). But whether or not the words themselves are related, their meaning definitely is.
In all three traditions, the division of scripture into numbered chapters and verses came much later. No rabbi of the Talmud, for example, ever quotes scripture with reference to a chapter or verse number. Likewise, in the “New Testament,” no verses of the “Old Testament” are quoted by chapter/number. In both Talmud and NT, they’re always quoted using the actual words involved (or a few of them).
Mohammed originally believed that he was continuing in the Jewish prophetic tradition; Muslims originally prayed facing Jerusalem! Only later was the orientation of Muslim prayer changed to Mecca (as recorded in the Kur’an itself):
“Fools among the people will say, ‘What has turned them from the Qibla [the direction they faced in prayer]…?’ Say, ‘To G-d belong both East and West…’. ” 
A Muslim commentary (one which preceded the state of Israel) on this verse states:
“In the early days, before they [i.e. the Muslims] were organized as a people [i.e. a distinct religious community], they followed as a symbol for their Qibla the sacred city of Jerusalem, sacred both to the Jews and Christians, the People of the Book [i.e. the Bible]. This symbolized their allegiance to the continuity of G-d’s revelation. When, despised and persecuted, they were turned out of Mecca and arrived at Medina, Mustafa [i.e. Mohammed] under divine direction began to organize his people as an Ummat, an independent people, with laws and rituals of their own. At that stage, the Ka’ba [in Mecca] was established as a Qibla’… “ 
He believed, though, that Jews and Christians had “corrupted” the Scriptures (or the proper understanding of them) which he was now correcting by a new series of Divine revelations in the form of public “readings.”
Of course, the “Kur’an” isn’t simply a “copy” of the Torah. Rather, I’d liken it to an inspired artist using pre-existing forms and materials to create something original. Rembrandt, for example, didn’t invent either the materials or the artistic form of oil-painting. Rather, he used existing materials, techniques and forms in a uniquely inspired and innovative way. Likewise, saying that the Kur’an borrows certain aspects of Torah — especially aspects in place by Mohammed’s time — takes nothing away from its originality or compelling spiritual force.
The public reading of Torah, in the form current by the end of the 2nd Temple period, served as the template for later traditions.