Learning Torah all night:[1]

It is customary for those gathered [and especially the father of the child] to stay awake throughout the night prior[2] to the Bris Mila studying Torah in the house of the newborn.[3] The Torah is to be studied [at least for some of the time] near the newborn child. At the very least, one should study Torah in the home until midnight.[4] It is proper for those present to be Torah scholars and men of stature, who will speak words of Torah and retell stories of Tzadikim throughout the night.[5]

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[1] Beis Lechem Yehuda 265:1; Zohar 1:93 that there was an individual who gathered his friends to study Torah together throughout the entire night, on the night prior to the Bris; Shlah Miseches Chulin; Or Tzadikim Hilchos Mila; Halachos Ketanos 2:169 that so is the custom of Jerusalem; Siddur Yaavetz; Migdal Oz of Yaavetz; Kuntrus Chanoch Lanar p. 7 that so was done by the Wacht Nacht of the Rebbe Rashab, that his uncles who are the holy sons of the Tzemach Tzedek, and dozens of Chassidim, stayed awake throughout the night learning Torah and reciting sections of Scripture and the Zohar; Shevach Habris 12:3; Kovetz Minhagim; See Encyclopedia Talmudit Vol. 4 Erech Bris Mila p. 247 footnote 22

[2] The custom of the world is to do so only on the night prior to the circumcision. However in previous times in Eretz Yisrael it was customary to do so on all nights prior to the circumcision.

[3] The reason: The learning of Torah at this time has ability a draw down a great holiness onto the child and to all those who are present studying. [Siddur Yavetz ibid]

[4] Zecher Dovid Mamar 1:30 in name of Mavor Yabok; Bris Avos Kuntrus Leil Shimurim 12 in name of Divrei Habris “It is customary to learn in the home of the newborn child up until midnight.”

[5] Zochar Habris 3:18; Otzer Habris 3:7-11