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It is a positive command of the Torah for every single Jew to count seven weeks’ worth of days beginning from when the Omer was offered. [In the times of the Temple the Omer was offered on the 16th of Nissan, the second day of Pesach. The Omer offering consisted of the new harvest of barley and was brought as a Mincha offering.]
In the Diaspora: The Mitzvah of counting the Omer Biblically applies in both Eretz Yisrael and the Diaspora.
During exile is the Mitzvah of Biblical or Rabbinical status? It is disputed as to whether the obligation of counting the Omer during the times of exile, when we no longer have a Temple and no longer bring the Omer offering, is of Biblical requirement or is a Rabbinical injunction. Practically, the main Halachic opinion follows like those who hold that the Mitzvah today is of Rabbinical status. Nevertheless, despite its Rabbinical nature, the institution of the Sages follow the same laws as the Biblical command and there is thus no practical Halachic differences between them, other than those cases to be mentioned.
Is the Biblical Mitzvah dependent on the Omer or on the Shtei Halechem?
The Poskim rule that the Biblical Mitzvah of counting the Omer is also dependent on the Shtei Halechem offering which is brought on Shavuos.
If Moshiach arrives in middle of Sefira, does the Mitzvah become a Biblical obligation?
Yes. Although the Mitzvah of Sefiras Haomer is dependent on the brining of the Omer, the Poskim rule that it is also dependent on the Shtei Halechem offering which is brought on Shavuos, and hence once Moshiach comes and one is able to offer the Shtei Halechem, the Sefira from that day and onwards becomes Biblical. [Some suggest that even the past days of Sefira that year, prior to Moshiach’s arrival, retroactively become Biblical once he arrives. The Rebbe however only suggests that the future days will become Biblical.]
 Admur 489/1
 Admur ibid; Michaber 489/1; Menachos 65b; This is coming to exclude the idea that only the Beis Din is obligated in the counting. It however is not coming to include women within the obligation, as women are explicitly exempt from this Mitzvah as explained in 489/2.
 Rambam Temidim Umusafim 7/11
 Admur 489/2; Rambam Temidim Umusafim 7/24; Ravayah 526
 Admur 489/2
 The dispute: [According to some opinions] this [Biblical] Mitzvah of counting the Omer [Biblically] applies even [during exile] when we do not have a Temple. [1st opinion in Admur ibid; Rambam Temidim Umusafim 7/24; Ravayah 526; Chinuch 306] However, other opinions rule that today in exile, when we no longer have a Temple or bring the Omer offering, the Mitzvah of Sefira is no longer at all a Biblical command and is rather a Rabbinical obligation that was instituted in memory of the Temple. [2nd opinion in Admur ibid; Tosafus Menachos 66a; Rosh Pesachim 10/40; Ran Pesachim 28a; See Beis Halevi 1/39] Other opinions rule it is no longer Biblical because we do not have the Temple and cannot bring the Shtei Halechem on Shavuos, [and not because of the Omer offering]. [Likkutei Sichos 38/10 [printed in Toras Menachem 3/4-12] and Toras Menachem 5751 3/95; Radbaz 1327; Abudarham Seder Tefilos Shaar 3; Maharil Seder Birchas Haomer; Beis Yitzchak Y.D. 2/19; Sefer Hamachria 18; Nimukei Orach Chaim in end of Sefer; Darkei Chaim Veshalom 625; Minchas Elazar 5/1; Piskeiy Teshuvoas 489/10] The Ran ibid explains an alternative reason for why the Sages established the Mitzvah of Sefira today in exile is because even originally the Mitzvah of Sefira was not relevant to the Omer or Temple, as the Jewish people first began counting the Omer in anticipation of receiving the Torah, and hence today too we count the Omer in anticipation for Shavuos. Nevertheless, this is merely a Midrash, as the main reason for the institution was for commemoration of the Temple. [Ran ibid]
 Admur ibid; Beis Yosef 489; implication of Michaber 489/6; M”B 489/14; Kaf Hachaim 489/4
 These differences include a) If one counted the days and not the weeks. [489/6] b) Counting by Bein Hashmashos. [489/12-13]
 Likkutei Sichos 38/10 [printed in Toras Menachem 3/4-12] and Toras Menachem 5751 3/95; Radbaz 1327; Abudarham Seder Tefilos Shaar 3; Maharil Seder Birchas Haomer; Beis Yitzchak Y.D. 2/19; Sefer Hamachria 18; Nimukei Orach Chaim in end of Sefer; Darkei Chaim Veshalom 625; Minchas Elazar 5/1; Piskeiy Teshuvoas 489/10
 Likkutei Sichos 38/10 [printed in Toras Menachem 3/4-12] and Toras Menachem 5751 3/95
 See previous footnotes
 Munkatcher ibid