The history of the reading of the weekly Parsha in Shul [i.e. Kerias Hatorah]:
What should be read-who established this and when? The Sages of the Mishneh and Gemara chose the exact portion of the Torah that is read on each holiday. However, according to most approaches, they did not establish what portions should be read on each Shabbos, and hence in previous times two different customs existed regarding this matter.
The customs of the weekly Parsha-annual cycle versus triennial cycle: The widespread custom even in previous times was to read one Parsha per week and arrange to finish the entire Chumash annually. [Possibly, this tradition started already in the times of Moshe Rabbeinu.] There were communities, however, who were accustomed to follow a triennial cycle and hence finish the entire Chumash every three years. This was the custom of Jewry in Eretz Yisrael during Talmudic times. Each Parsha was thus split up to three parts, with each part being read in one week. This minority custom became extinct several generations ago and hence the custom amongst all Jewry dating back many generations is to read one Parsha per week and complete the reading of the entire Torah annually.
The order of the annual cycle: The set order of these Parshiyos, when to begin Bereishis, when to finish Zos Habracha, and which Parshiyos to connect, is based on the order suggested by Rav Sadia Gaon, as recorded in his Siddur Rasag.
Prohibition to swerve from the annual cycle, and its esoteric connection: Practically, the weekly Torah portion for each Shabbos has the full backing of all Minhagei Yisrael, Jewish customs, and it is hence forbidden to switch the Parshiyos and change the custom. Furthermore, the Parsha of each week has a special connection with the events and times of that week, and it is all arranged with Divine providence.
 Admur 282:1; 13; 488:5; M”A 135:1; Rambam Tefila 12:1 “Moshe”; Rif Megillah 4; Bava Kama 82a that the prophets in the times of Moshe established it, See Kesef Mishneh ibid; Yerushalmi Megillah 4:1 “Moshe”; Miseches Sofrim 10:1; Mishneh Megillah 31a regarding Yom Tov; See P”M 135 A”A 1
 P”M 135 A”A 1 that so is implied from Setimas Haposkim in 135 and 685; Tosafus Megillah 17b
Other opinions: Some Poskim rule that Kerias Hatorah on Shabbos is a Biblical obligation. [Bach 685 and that so is opinion of Rashi, brought and negated in P”M ibid]
 Admur ibid; Megillah ibid
 See Bach 685 “However, Moshe did not establish the order of what should be read on Shabbos until Ezra came along.” See Rambam Tefilah 13:1-2 that the weekly Torah portion read on Shabbos is based on custom and not law, although Ezra established that certain Parshiyos are to be read at certain times, as brought in the Michaber 428 and 685. The fact that Ezra gave rules regarding certain Parshiyos and when they are to be read seems to prove that there was never an order of one set Parsha per week, and rather every community could choose their order, so long as they follow the rules of Ezra. See Piskeiy Teshuvos 135:1
Other opinions: Some Poskim write that Moshe established which Parsha is to be read each Shabbos for all the Parshiyos of the Torah. [See Admur 282:13 “The reading of the Parsha every Shabbos was a Takana of Moshe Rabbeinu A”H”; See Zohar Vayakhel p. 206b “It is forbidden to stop in a Parsha that Moshe did not stop, and it is forbidden to read a different weeks Parsha on that Shabbos.” M”A 282:1 explains that the intent of the Zohar is to say that one may not read another weeks Parsha on Shabbos, and each Shabbos must have its designated Parsha read. Thus, we see that Moshe himself handed us a tradition of the start and end of each weekly Parsha; Yeish Sechar Dinei Kerias Hatorah 6 [1600’s] writes that Ezra Hasofer established all seven Aliyos of each Parsha based on a tradition dating all the way back to Moshe; Aruch Hashulchan 282:2 “Moshe Rabbeinu established which Sedra should be read each Shabbos.”; Shulchan Hatahor 135:4 that even the double Parshiyos are a tradition of Moshe from Sinai; See Piskeiy Teshuvos 135:1]
 Rambam ibid
 Hisvadyus 1984 2:1082; See the Pizmon of Sisu Visimchu that is recited after Kerias Hatorah in which we state that Moshe rejoiced on Simchas Torah, and it should be accepted literally, as a) Moshe Rabbeinu established the reading of the Torah during the weeks; b) It is difficult to assume that some person in a later generation established it, as why would Klal Yisrael suddenly follow his custom. Rather one must say it was already followed by Moshe Rabbeinu as a custom. [Hisvadyos ibid]
 Megillah 29b; Rambam Tefilah 13:1-2; Testimony of Rav Binyamon Tudela regarding the Egyptian community; Manuscripts in Cairo Geniza; Some sources state that they split the Torah to 155 Parshiyos. Others say they split it to 167 Parshiyos, while others say it was split to 141 Parshiyos.
 Megillah 29b
 Some historians claim that this custom became extinct in the Geonic period. However, see Rambam ibid from whom it is clear that it existed in his times, and so in truth we found historical manuscripts from the Geniza Kahir which clearly prove that in the Rambam’s time period there still existed Egyptian Jewish communities, known as Shamites, who followed the triennial cycle of Eretz Yisrael. According to the manuscript, pressure was placed on these communities to cease their practice although they did not heed the request and continued their practice, with some compromise at least until the 1600’s.
 Aruch Hashulchan 282:3
 Siddur Rasag Hilchos Kerias Hatorah, printed by Mikitzei Nirdamim p. 363
 Zohar Parshas Vayakhel p. 369 [206b], brought in M”A 282:1, “One may not read the Parsha of another week, this week.”; M”A 282:1 in explanation of Zohar ibid; Aruch Hashulchan ibid “Chalila to switch one Parsha for another”
 See Divrei Torah 9:93; Shlah Vayeishev; Rebbe in various Sichos