Gemara

(R. Yehudah bar Bizna): When Moshe slackened from circumcising his son, the damaging angels Af and Chemah swallowed his entire body, except for his legs.

Immediately, Tziporah circumcised her son with a rock, and the angels left Moshe alone.

Avodah Zarah 26b (Beraisa - Chachamim): A Nochri may circumcise a Yisrael, only if others are there.

27a (Beraisa - Rebbi): Circumcision by a Nochri is invalid - "V'Atah Es Brisi Tishmor".

Question: What is the source that circumcision by a Nochri is invalid?

Answer #1 (Rav): We learn from "V'Atah Es Brisi Tishmor".

Answer #2 (R. Yochanan): We learn from "Himol Yimol" (one who is circumcised may circumcise).

These answers differ about a Yisraelis. She is not commanded about circumcision, but she is considered circumcised.

Question: Surely a woman may circumcise! "Va'Tikach Tziporah Tzor..."!

Answer #1: We may read this 'va'Tukach' (she told someone else to take the rock and circumcise).

Question: It says "Va'Tichros" (she cut)!

Answer: We may read this 'va'Tichras' (she told someone else to cut).

Answer #2: She started the circumcision, and Moshe finished it.

Kidushin 29a (Mishnah): Regarding any Mitzvah of the son on the father, men are obligated, and women are exempt;

(Rav Yehudah): This refers to a Mitzvah that the father must do to the son.

(Beraisa): A man is obligated to circumcise his son...

Question: What is the source of this?

Answer: "Avraham circumcised his son (like Hash-m commanded him)" teaches that a man is commanded to circumcise his son.

Question: What is the source that women are exempt?

Answer: "Like Hash-m commanded him" - not her.

Question: That was said regarding Avraham and Sarah. What is the source for all generations?

Answer (Beraisa - Tana d'Vei R. Yishmael): Wherever it says "command", this teaches that one must be zealous, and that it applies for all generations.

Rishonim

Rif and Rosh (Shabbos 56a and 19:11): The Halachah follows R. Yochanan against Rav. Therefore, if the only one around who knows how to circumcise is a woman, this is fine.

Rambam (Hilchos Milah 2:1): Everyone is Kosher to circumcise. Even a slave, woman or child can circumcise where there is no man

Tosfos (Avodah Zarah 27a DH Ishah): Normally, the Halachah follows R. Yochanan against Rav. Here the Halachah follows Rav, because Rebbi's Beraisa supports him. BaHaG says that a woman can circumcise. We expound "(like Hash-m commanded) him", not her. Rav must explain this to teach that she need not ask others to circumcise her son.

Ha'Makneh (Kidushin 29a DH uv'Zeh): Perhaps Rav's source that women cannot circumcise is "him"! It seems that "him" would exempt only from the special obligation on the father. Rav must hold that since Milah does not apply to her, the Mitzvah does not apply to her at all.

Poskim

Shulchan Aruch (YD 264:1): Everyone is Kosher to circumcise, even a slave, woman or child. If an adult Yisrael knows how to circumcise, he has precedence.

Rema: Some say that a woman should not circumcise. The custom is to seek a man.

Beis Yosef (DH ha'Kol): The Kolbo cites SMaK, who says that a woman may not circumcise. Perhaps this is l'Chatchilah.

Rebuttal (Bach DH v'Kosvu): SMaK is stringent like Tosfos to equate a woman to a Nochri. If necessary, we postpone the Milah until we find a male Yisrael who knows how to circumcise. The Mordechai (Shabbos 468 p. 166, DH v'Ha) concludes like the SMaK; we should be stringent.

Question (Shach 2 and Gra 6): What does it mean 'the custom is to seek a man'? The Shulchan Aruch agrees that we prefer a man! If the Rema forbids a woman to circumcise even when no man is available, like the Bach, it is unreasonable to discuss a custom for a situation that is very rare!

Ha'Makneh (ibid.): The Rema holds that the Rambam and Shulchan Aruch do not allow a woman to bless on Milah, just like any Mitzvas Aseh sheha'Zeman Grama. The Rema (589) allows them to bless. The custom is to seek a man due to concern for Rav's opinion.

Pischei Teshuvah (3): Beis Yakov (104) says that if a woman (or anyone Pasul for Milah) began to circumcise, and a man who knows how to circumcise came, she must stop immediately. He finishes the Milah, and it is Kosher.

Aruch ha'Shulchan (8): Even if a Pasul cut the entire foreskin and a Kosher did Pri'ah (exposing the flesh under the foreskin) it is Kosher, for Milah without Pri'ah is as if no Milah was done. Therefore, it is as if the Pasul merely started. Even if this is wrong, we can rely on the Rambam (who is Machshir women) not to require another Milah.

Teshuvas Chelkas Yakov (OC 112 DH b'Shulei): The Shulchan Aruch (266:14) requires one who circumcises on Shabbos to do Pri'ah; the Rema permits someone else to do Pri'ah. They argue about whether or not the Mitzvah of Milah is only the completion, or if even the initial cutting is included. They argue like the two answers in the Gemara regarding Tziporah. The first answer said that she told someone else to circumcise, for the entire Milah is part of the Mitzvah, so a woman cannot even begin. The latter answer holds that the Mitzvah is only the end, so she was able to begin.

Teshuvas Beis She'arim (YD 350 DH ul'Fi): According to the text that Avraham was commanded about Pri'ah (but did not do so to himself, for his foreskin was already flattened by many Bi'os), how was he able to circumcise Yitzchak? We hold that a woman is Kosher for Milah. It is as if she was circumcised. Similarly, it is as if Pri'ah was done to Avraham!

Teshuvas Beis She'arim (YD 358 DH v'Al): A woman who converted can be a Shali'ach to convert her son if he was already circumcised. If not, she cannot, for Milah does not apply to her. Even the opinion that says that she is Kosher for Milah says so only when she has a husband, like Tziporah.

Magen Avraham (187:3): Some say that a woman says 'v'Al Torascha she'Limadtanu' in Birkas ha'Mazon because she and her husband are like one body. Why does Tosfos (Berachos 20b DH Nashim) say that women should not mention Bris in Birkas ha'Mazon? All agree that it is as if she is circumcised, so she can say ''v'Al Brischa she'Chosamta bi'Vsarenu'!