The 15th of Av:
The Mishneh states: Rebbe Shimon Ben Gamliel said “There were not days of joy for the Jewish people like the 15th of Av and Yom Kippur.” This day became a day of Shidduchim, where men would pick for themselves brides. The potential brides would all go out with white clothing and tell the bachelors to choose for themselves a girl of good character and lineage and not look at matters of vanity, such as beauty.
A. The occurrences:
The Gemara lists several opinions regarding matters that occurred on the 15th of Av, which is the cause for its great celebratory status. These reasons are not contradictory of each other, and in truth all occurred on this day. Each Sage mentions the matter which he received as a tradition from his teacher.
- Rav Yehuda stated in the name of Shmuel that on this day the Shevatim were allowed to intermarry with each other. [In the first generation that entered Eretz Yisrael, it was prohibited for the member of one tribe to marry the member of another tribe, due to the conflicts this could potentially cause in the inheritance of portions of land in the region of one tribe to another tribe. This prohibition only applied in the first generation and was officially revoked on the 15th of Av.]
- Rav Yosef and Rav Yochanon stated that on this day the tribe of Binyamon was permitted to marry. [During the episode of Pilegesh Begiva, the tribe of Binyamon was nearly annihilated and could not marry, due to all its daughters being killed, and due to the remainder of the Jewish tribes having sworn not to give them any of their daughters as wives. There were 600 men of Binyamon remaining and no women available to be given to them in marriage. While 400 girls were obtained from the town of Yaveish Gilaad, they still needed to obtain 200 more. On the 15th of Av, when the girls went to dance in Shilo, the 200 remaining men of Binyoman came and took for themselves wives. Accordingly, this day became a national event of salvation for the tribe of Binyoman, saving them from annihilation.]
- Raba Bar Bar Chana said it is the day that the generation of the Midbar ceased to die. [Each year on Tisha B’av, throughout the 40 years in the desert, the men would dig for themselves a grave and lie in it throughout Tisha B’av, expecting their ultimate death that day. In the 40th year, everyone got up alive in their grave. They wondered how this happened and that perhaps they made a mistake in the calendar. It was only until the 15th of the month of Av, when they saw the full moon, that they realized there was no mistake, and the Divine decree had been rescinded.]
- Ula says it is the day that Hosheia Ben Ala removed the blockades that were placed by Yeravam Ben Nevat that prevented the Jewish people from visiting the Temple in Jerusalem.
- Rav Masna says that on this day the corpses of Beitar were permitted to be buried.
- Raba and Rav Yosef said that on this day the cutting of wood for the Temple ceased.
The true reason behind its greatness:
In the writings of the Arizal it is explained that the greatness of the 15th of Av is that on this day we experience a full moon. The full moon represents a high spiritual level. Now, although in truth every month contains a full moon, the full moon of this month is unique being that it comes after the great descent of Tisha B’av. Hence in comparison to the great descent experienced some days prior, the elevation now experienced is so much greater, that it became a day fit for celebration.
B. Adding in Torah learning at night:
From the 15th of Av and onwards the strength of the sun is diminished. This means that there are less daylight hours in the day, and consequently more nighttime hours. The nighttime is an auspicious time for Torah learning. Accordingly, starting from the 15th of Av, one who increases in Torah learning at night, using the newly added night hours for spiritual service, will have days added to his life. One who doesn’t, will have his days shortened. [Learning at night contains special Segulo/powers. The quality of one’s learning is greater, as one is able to feel more connected to Hashem in one’s learning. One learns majority of his wisdom at night and is able to merit Keser Torah through learning at night.]
D. Wishing Kesiva Vechasima Tova
 Taanis 26b
 Taanis 30b; Bava Basra 121a-b
 Rashi Bava Basra ibid
 See Rashi ibid
 See Shoftim 20-21
 Rashi ibid
 Mamar Lo Hoyu Yomim Tovim 5735; Rebbe Rashab Mamar Nachamu 5670
 Peri Eitz Chaim Shaar Chag Hashavuos
 Taanis 31a; Bava Basra 121a-b
 Admur Hilchos Talmud Torah 4:8; Rama 245:23
 Likkutei Sichos 34 p. 43
 Admur Hilchos Talmud Torah 4:8
 Siddur Admur; Michaber 131:6; Taz 421:1; Levush 131:6; Peri Chadash 131; Hayom Yom 14th Menachem Av
 Siddur Admur; Levush ibid; Peri Chadash ibid; Hayom Yom 14th Menachem Av
 “You shall be written in and sealed for a good year”. This is hinted to in the words “Vayishalu Ish Lireieihu Leshalom” which is also an acronym for Elul [Likkutei Mahrich]
 Custom to begin from the 15th of Av: Shaar Yissacher and Darkei Chaim Veshalom [684; Munkatch] write that one is to begin wishing each other Leshana Tova starting from the 15th of Av. The Rebbe states that it is a Minhag Yisrael to do so. [Hisvadyos 1991 Vol. 4 p. 179; printed in Shulchan Menachem 3:72] The Gematria of “Chamisha Asar Beav” is the same as “Kesiva Vechasima Tova”. [Darkei Chaim Veshalom ibid; Hisvadyos ibid] This is also hinted to in the acronym for the Mazal of Menachem Av which is “Aryeh” and stands for Elul; R”H; Yom Kippur; Hoshanah Raba, as in the month of Av is hinted to the completion of the Piska Tova of Hoshanah Raba. [Rebbe ibid] This blessing is found in letters of the Rebbe Rayatz and Rebbe written in the month of Menachem Av. [Otzer p. 11-13]