Gemara

(Bar Kapara): "This is the law of animals and birds (and... fish)" mentions birds between animals and fish.

Birds are Hukshu (written next to, and thereby equated) to fish (which do not need any Shechitah), and to animals (in which both Simanim must be slaughtered). We must say that one Siman must be slaughtered.

Question: What is the source that fish need not be slaughtered?

Suggestion: "Ha'Tzon u'Vakar Yishachet... Degei ha'Yam Ye'asef" teaches that Asifah (gathering) suffices for fish.

Question: If so, we should say the same about Slav (the fowl given in the Midbar), about which it says "va'Ya'asfu Es ha'Slav"! However, we said that birds require Shechitah!

Answer: Indeed, we learn from "ha'Tzon...";

"Ha'Tzon..." mentions Shechitah of animals, and correspondingly, gathering fish. "Va'Ya'asfu Es ha'Slav" does not correlate this to Shechitah, so we do not learn from it.

16b (Beraisa - R. Yishmael): "When Hash-m will widen your border... and you will say 'I want to eat meat'" permits Basar Ta'avah (eating animals like Chulin, i.e. not a Shelamim);

In the Midbar, Basar Ta'avah was forbidden. When Yisrael entered Eretz Yisrael, it was permitted.

17a (Beraisa - R. Akiva): "When you will be far from (the Mikdash) and you will slaughter your cattle and flock" forbids an animal killed through Nechirah (tearing the Simanim lengthwise);

In the Midbar, Nechirah was permitted. When Yisrael entered Eretz Yisrael, it was forbidden.

R. Akiva holds that Basar Ta'avah was never forbidden. R. Yishmael holds that Nechirah was never permitted.

Question: According to R. Yishmael we understand "could enough Behemos be slaughtered for them?" However, according to R. Akiva, why does it say "slaughtered"? It should say 'Yinacher'!

Answer: Since Nechirah was permitted, it was like Shechitah.

Kerisus 21b (Beraisa): We exclude (blood of) fish and Chagavim (locusts or grasshoppers), for they are wholly permitted.

This means that they are permitted without Shechitah.

Shabbos 90b - Question: Why do Chachamim hold that people do not save a Tamei Chagav for children to play with it?

Answer #1: They are afraid lest the child eat it.

Objection: If so, they should be concerned also about Tahor Chagavim!

Rav Kahana put Shushiva (a Tahor Chagav) in front of his mouth.

Rav: Remove it, lest people say that you ate it and transgressed Bal Teshaktzu (doing something disgusting)!

Answer #2: They are afraid lest it die, and the child will eat it.

Avodah Zarah 38a (Rav Berona): If a Nochri made a fire in a swamp, the Chagavim (that became roasted) are forbidden.

The case is, one cannot tell whether they are Tahor species. The same applies if a Yisrael made the fire. A case occurred with a Nochri, so the Halachah was said regarding a Nochri.

40b (Beraisa): Chagavim from a Nochri's storehouse, pile or boat are permitted;

Those sold in the market in front of the vendor are forbidden, because they sprinkle wine on them.

Rishonim

Rambam (Hilchos Shechitah 1:3): Fish and Chagavim need not be slaughtered. Gathering them permits them. "Ha'Tzon u'Vakar Yishachet... Degei ha'Yam Ye'asef" teaches that gathering fish is like Shechitah of cattle or flock. Regarding Chagavim it says "Osef ha'Chasil" - they are permitted through mere gathering.

Rashi (Chulin 66a DH u'Mar, citing Bahag): It says 'this is the law of animals and birds v'Chol Nefesh Chayah ha'Romeses ba'Mayim v'Chol Nefesh ha'Shoretzes Al ha'Aretz. The last words refer to Chagavim; they are not included in 'birds'. They are mentioned after fish to teach that they do not need Shechitah, just like fish do not. (This is brought also in Hagahos Maimoniyos 2 and Semag Aseh 62.)

Magid Mishneh: The Rashba challenged the Rambam, for the Gemara says that we learn from "Asifah" only when it is written corresponding to Shechitah of other species! He concluded that there is no clear proof to permit Chagavim (without Shechitah). Perhaps the Rambam's words are a mere Asmachta. Rashi brought the Ge'onim's proof. Several Gemaros prove that they are permitted without Shechitah.

Kesef Mishneh: We can say that really, it suffices to say Asifah even not corresponding to Shechitah. This was not enough for fish, for one might have thought they are written in the verse with animals to equate them and require Shechitah. Since we needed a verse to obligate Shechitah for Chayos and birds, why do we need a verse to exempt fish and Chagavim? Initially, we thought that only Behemos require Shechitah, for the Torah explicitly says so. Once we find that also other things require Shechitah, we need a source also for fish and Chagavim.

Beis Yosef (YD 13 DH Dagim): Rather, Shechitah was never written regarding Chagavim. If the Torah did not explicitly say that a certain matter is forbidden without Shechitah, it is permitted. The Gemara asked about fish, for they are written together with Behemos.

Kesef Mishneh: We should require fish and Chagavim to be gathered while alive! If we could say so, this would be true. However, one cannot eat them without gathering them. Even if he did not gather them with his hands, they are gathered in his mouth. If so, the Torah did not command about this.

Lechem Mishneh: Hagahos Maimoniyos says that also the Rambam learns from the Hekesh to fish. The Rambam connotes otherwise. He did not mention the Hekesh! I say that the Rambam found the Gemara difficult. Why did it challenge the source for fish from the verse about Slav? Slav is different, for it is equated to animals! The Gemara answered that since "Asifah" for fish was said corresponding to Shechitah of animals, this overrides the Hekesh. Regarding Chagavim, there is no Hekesh, so we learn from Asifah.

Rambam (ibid.): Therefore, if they died by themselves in water they are permitted, and one may eat them alive.

Hagahos Maimoniyos (3): A Tosefta in Terumos permits eating Chagavim live or slaughtered. Actually, one may not eat them (live), for Bal Teshaktzu applies. Rather, if one ate them alive, Ever Min ha'Chai does not apply, for they need not be slaughtered.

Magid Mishneh: There is no proof from Shabbos. Some say that Shushiva is a Tamei species. Perhaps this is why Rav told him to remove it! Alternatively, the Rambam teaches only that they are permitted regarding Ever Min ha'Chai.

Note: Rav told him 'lest people say that you transgressed Bal Teshaktzu.' The Magid Mishneh holds that this could refer to "Al Teshaktzu... b'Chol ha'Sheretz..." (Vayikra 11:43)!

Radvaz: The Rambam permits eating limbs from them. Bal Teshaktzu applies to eating them whole. Why does the Rambam permit if they died in water? The same applies if they died outside the water! It seems that the Chidush is even if they died by themselves, and he did not gather them, they are permitted.

Tosfos (Shabbos 90b DH d'Lo): The Rashba derived from here that Chagavim do not require Shechitah. Rav was not concerned lest people say that he eats without Shechitah. Also, Chachamim are concerned only for eating a dead Tamei Chagav, but not for a dead Tahor one. Even though one may not eat it live due to Bal Teshaktzu, one may cut off a limb and eat it through washing it (so it will not be disgusting).

Tosfos (Chulin 17a DH Rebbi): What is the source to say that R. Akiva holds that Basar Ta'avah was never forbidden? Perhaps he and R. Yishmael taught different matters, and they do not argue! One could say that if R. Akiva held that Basar Ta'avah was forbidden, how was Nechirah permitted? He agrees that Kodshim require Shechitah. Nechirah is not for Chayos, for the verse from which we derive the Heter of Nechirah discusses Behemos. However, perhaps the verse discusses blemished animals!

R. Akiva Eiger: Tosfos did not answer his question. I ask, how can we learn that fish do not need Shechitah from "... Degei ha'Yam Ye'asef"? According to R. Akiva, in the Midbar Nechirah was permitted. This is why gathering sufficed for fish. Now that Nechirah is forbidden, perhaps also fish require Shechitah! This is not difficult. The verse mentions Shechitah of Behemos, even though Nechirah sufficed, for Nechirah in the Simanim was like our Shechitah. Therefore, the verse teaches that fish did not require even Nechirah. Therefore, we need not slaughter fish. This proof is good for R. Akiva and R. Yishmael. This answers Tosfos' question. If R. Akiva and R. Yishmael agreed with each other, we would have no source that fish do not require Shechitah. In the Midbar they could eat Behemos only through being Makdish them, and they required proper Shechitah, but Nechirah sufficed for fish, just like for other animals! Now that Nechirah is forbidden, also fish are forbidden (without Shechitah)! Rather, we must say that they argue.

Poskim

Shulchan Aruch (YD 13:1): A Behemah, Chayah or bird requires Shechitah. Fish and Chagavim do not require Shechitah.

Rema: One may eat them dead or cut a limb from them and eat it, but one may not eat them alive, due to Bal Teshaktzu.

Beis Yosef (DH v'Chasav): Also Ba'al ha'Itur says that one may eat them alive, unlike Rav Sadya Gaon who forbids them alive. R. Nisim supports this from the Tosefta (Terumos 9:6), which says that one may eat fish and Chagavim alive or dead, without concern. Also the Rashba brought the Tosefta. I say that Rav Sadya Gaon did not forbid due to Ever Min ha'Chai, rather, due to Bal Teshaktzu, like it says in Shabbos.