Gemara

130a (Mishnah): (One who slaughters a Chulin Behemah is commanded to give Zero'a, Lechayim and Kevah (the upper two bones of the foreleg, the jaw and stomach) to a Kohen). The Mitzvah applies in Eretz Yisrael and Chutz la'Aretz, whether or not the Mikdash stands.

131b: Ula would give Matanos Kehunah to a Kohenes.

Rav Kahana (a Yisrael married to a Kohenes) received and ate Matanos due to his wife, and so did Rav Papa, Rav Yemar, and Rav Idi bar Avin.

(Ravina): The Halachah follows Ula.

132b (Rav Nachman, citing R. Acha bar Chanina): A Kohen who slaughters is exempt for two or three weeks. After this, he must give the Matanos to another Kohen.

(Rav Chisda): A Kohen who does not separate the Matanos is excommunicated.

(Rabah bar Rav Shila): The Shochtim of Hutzal have been excommunicated due to Rav Chisda's decree for 22 years.

Rava would give the entire thigh to a Kohen (from Shochtim who refused to give Matanos). Rav Nachman bar Yitzchak would give the Shochet's garment to a Kohen.

(Rabah bar bar Chanah citing R. Yochanan): One may not eat from an animal before Matanos are separated.

(Rabah bar bar Chanah citing R. Yochanan): Eating from an animal before Matanos are separated is like eating Tevel.

The Halachah does not follow Rabah bar bar Chanah.

136a (Beraisa - R. Ila'i): Matanos do not apply in Chutz la'Aretz. The same applies to Reishis ha'Gez.

(Rava): He exempts because it says "Nesinah" regarding Reishis ha'Gez, like it says regarding Terumah.

Question (Abaye): If so, Reishis ha'Gez should be Tovel (forbid benefit from shearings before it is separated), just like Terumah is Tovel!

Answer (Rava): "Reishis Gez Tzoncha Titen Lo" - before it is separated, it has no effect (to forbid).

136b (Rav Nachman bar Yitzchak): Nowadays, everyone conducts according to the lenient opinion of R. Ila'i regarding Reishis ha'Gez;

(Beraisa - R. Ila'i): Reishis ha'Gez applies only in Eretz Yisrael.

138b (Mishnah): Sending the mother bird applies in Eretz Yisrael and in Chutz la'Aretz, whether or not the Mikdash stands.

(R. Avin or R. Meisha): In all these Perakim (from the 10th onwards) it says that the Mitzvah applies in Eretz Yisrael and in Chutz la'Aretz. There was no need to teach this, except for Reishis ha'Gez (and Perek ha'Zero'a, regarding Matanos)

There, it was necessary to say this, to exclude the opinion of R. Ila'i, who exempts in Chutz la'Aretz.

Shabbos 10b: Rav Chisda had two (sets of) Matanos, and offered them to whoever would tell him a teaching of Rav that he had not yet heard.

Yevamos 63b: In Bavel, they decreed that Yisre'elim may not eat slaughtered meat, because Yisrael did not give Matanos to Kohanim.

Rishonim

Rambam (Hilchos Bikurim 9:1): It is a Mitzvas Aseh for anyone who slaughters a Tahor Behemah to give the Matanos, i.e. the foreleg, jaw and stomach, to a Kohen. This applies always, whether or not the Mikdash stands, and everywhere, in Eretz Yisrael and Chutz la'Aretz.

Rosh (11:1): Rashi says that just like we follow R. Ilai regarding Reishis ha'Gez, the same applies to Matanos. There is one reason for both of them. We learn from a Gezeirah Shavah "Nesinah-Nesinah" from Terumah. Even though all the Amora'im held that Matanos apply in Chutz la'Aretz, and they used to give also Reishis ha'Gez, later generations retracted to rule like R. Ila'i. Rav Nachman bar Yitzchak himself, in his youth, used to give to a Kohen the garment of a Shochet who did not give Matanos! The Rambam and Rabbeinu Meir me'Rotenburg hold that they apply nowadays. This is not the custom.

Rosh (11:3): Rav Nachman bar Yitzchak taught that nowadays people follow R. Ila'i regarding Reishis ha'Gez. I explained that the same applies to Matanos.

Rashi (136b DH k'Rebbi): (Nowadays, people conduct like R. Ila'i regarding Reishis ha'Gez, and) the same applies to Matanos.

Rashi (10b DH Havah): Even though Rav Chisda was in Chutz la'Aretz, we do not learn from here that Matanos apply nowadays. We conduct like R. Ila'i, who says that Reishis ha'Gez applies only in Eretz Yisrael, and also Matanos. In the days of Rav Chisda they did not yet follow R. Ilai. In the days of Rav Nachman bar Yitzchak, they followed R. Ilai regarding Reishis ha'Gez, but not regarding Matanos. Now we find that we follow him even regarding Matanos. Just like they did not protest when people started following him regarding Reishis ha'Gez in the days of Rav Nachman, we do not protest now that people follow him regarding Matanos.

Ran (46b DH Gemara): Rashi says that we conduct like R. Ila'i regarding Reishis ha'Gez, and also regarding Matanos. This is because we learn both of them from "Nesinah-Nesinah." This is difficult, for latter Rabanan fined people who did not give them! We say in Yevamos (63b) that Chaverim (a group that came to Bavel) decreed about meat, because Yisre'elim were not giving Matanos. This shows that they should be given! The Ba'al ha'Ma'or says that perhaps regarding eating, we do not rely on R. Ila'i, for R. Yochanan said that one who eats from an animal before Matanos are separated is like eating Tevel. Terumah is Tovel (forbids before it is separated) in Chutz la'Aretz. Even R. Ila'i agrees about this. Also this is wrong. Rather, the one who rules like R. Ila'i regarding Reishis ha'Gez rules like him also regarding Matanos. Rav Nachman did not rule like him. He said only that people conduct like him. Rather, people accepted R. Ila'i's opinion about Reishis ha'Gez, but not regarding Matanos. Rava said that R. Ila'i learns Reishis ha'Gez from "Nesinah- Nesinah" from Terumah. Abaye asked that if so, Reishis ha'Gez should be Tovel, just like Terumah is Tovel. He did not ask that Matanos should be Tovel! This is because it is clear that Matanos are unlike Terumah, even though R. Ila'i learns both of them from Terumah.

Rebuttal (Ma'adanei Yom Tov 11:10): There was no need to ask about Matanos, because we already learned above (132b) that Matanos are Tovel, just like Terumah!

Ran (ibid.): The Rambam holds like this, that Matanos apply everywhere, even though Reishis ha'Gez applies only in Eretz Yisrael.

Beis Yosef (YD 61 Sof DH veha'Ran): It seems that also the Rif holds like this. He did not bring R. Ila'i's teaching that Matanos do not apply in Chutz la'Aretz. Some say that the Rambam holds that Matanos apply in Chutz la'Aretz only in Mitzrayim, Bavel and other places close to Eretz Yisrael, for Terumah and Ma'aser apply there (mid'Rabanan).

Bach (16): Also the Rashba says that Chachamim enacted Matanos in Bavel and other places where Terumah and Ma'aser apply.

Rashi (138b DH Levad): The Gemara said that the only Mishnah that needed to teach 'in Eretz Yisrael and in Chutz la'Aretz' was regarding Reishis ha'Gez, to exclude the opinion of R. Ila'i, who exempts in Chutz la'Aretz. Really, it was needed also for Matanos, which R. Ila'i learns from Reishis ha'Gez. The Gemara mentioned only the first Mishnah for which it was needed.

Mordechai (736): Rashi was asked why people do not give Matanos nowadays. The Mishnah says that they always apply, and all the later Amora'im gave them! Rashi answered 'you are correct. We do not protest against one who gives. He will be blessed! However, people are lenient like R. Ila'i. Just like he exempts Reishis ha'Gez (in Chutz la'Aretz), he exempts Matanos. Whenever it says 'the custom is', not only we do not expound in the Shi'ur, rather, even we do not rule (unlike the custom). If one did (give), we do not return them. Rabbeinu Meir says that it was obvious that Matanos apply in Chutz la'Aretz. Even though the custom is like R. Ila'i, who exempts Reishis ha'Gez in Chutz la'Aretz, because he learns from Nesinah- Nesinah, and he exempts also Matanos for this reason, the custom is not like him regarding Matanos. This is why Rav Nachman did not say that people conduct like him regarding Matanos. Rav Nachman himself used to fine people who did not give them. One may be stringent for himself, but he may not fine others! Do not say that the custom about Matanos varies from place to place. 'Everyone conducts' connotes that everyone conducts like this, or at least in Rav Nachman's locale. We find many Amora'im who separated Matanos, but we do not find any Amora in Bavel who separated Reishis ha'Gez. The Gemara said that 'in Eretz Yisrael and in Chutz la'Aretz' was needed only for regarding Reishis ha'Gez, to exclude the opinion of R. Ila'i, who exempts. The Gemara should have said that it was needed for Matanos, which was taught before Reishis ha'Gez! Since it was taught there for a need, it was taught also regarding Reishis ha'Gez needlessly, since we already know from Matanos that the Halachah does not follow R. Ila'i, since there is one reason for both of them! Rav Hai Gaon said that one who does not give should be excommunicated, like Rav Chisda said.

Poskim

Shulchan Aruch (YD 61:21): Matanos apply to everywhere, in Eretz Yisrael and Chutz la'Aretz, whether or not the Mikdash stands.

Beis Yosef (DH u'Mah she'Chasav uv'Chol): The Gemara connotes like this. All the Amora'im held that Matanos apply in Chutz la'Aretz, even in their time, after the Churban.

Shulchan Aruch (ibid.): Some say that they do not apply in Chutz la'Aretz, and this is the custom.

Beis Yosef (DH u'Mah she'Chasav b'Shem): The Tur cited Rashi to say that Matanos do not apply nowadays, to hint that we should not ask that they were given in the days of the Amora'im. However, he should have mentioned that nowadays they do not apply in Chutz la'Aretz.

Beis Yosef (DH u'Mah she'Chasav v'Chen): The Tur said that Maharam wrote like Rashi. The Rosh wrote that Maharam holds like the Rambam!

Drishah (DH v'Chen): With difficulty, we can say that Maharam rules like the Rambam regarding Mitzrayim, Bavel and other places close to Eretz Yisrael. He holds that Rashi in the rest of Chutz la'Aretz.

Bach (16): Maharam wrote l'Halachah that Matanos apply to in Chutz la'Aretz, but he explicitly said that our custom is not to give them, like Rashi said.

Gra (YD 61:20): The primary opinion is the former, like Rabbeinu Meir.

Beis Hillel (3): Really, nowadays they do not apply in Eretz Yisrael. The Rema (331:2) said that some say that nowadays Terumah and Ma'aser apply in Eretz Yisrael, just the custom is unlike this. He should have commented here that the same applies to Matanos!

Rebuttal (Kuntres Acharon of Pri Chodosh): The Beis Hillel heard false testimony that the custom is not to give them in Eretz Yisrael. There, Yisre'elim slaughter Nochrim's animals for them, which is exempt. One who slaughters his own animal gives Matanos. I myself merited to fulfill this Mitzvah. Beis Hillel also erred about the Rema. In Siman 331 he means that the custom is to consider Terumah to be mid'Rabanan, so some leniencies apply. Chalilah to say that the custom is not to separate Terumah and Ma'aser!