Gemara

(Beraisa - R. Eliezer): One should request his needs and then pray (Shemonah Esre) - "Tefilah l'Ani Chi Ya'atof v'Lifnei Hash-m Yishpoch Sicho";

R. Yehoshua says, one should pray and then request his needs;

8a - Chachamim: One should request his needs in 'Shome'a Tefilah.'

(Rav Yehudah): The Halachah is, one may request his needs in 'Shome'a Tefilah.'

(Rav Yehudah brei d'Rav Shmuel bar Shilas): Even so, he may add requests in each of the (13 middle) blessings at the end.

(R. Chiya bar Ashi): Even though one may request his needs in 'Shome'a Tefilah', if someone in his family was sick, he requests for him in Birkas Refa'enu. If he needs income, he requests in Birkas ha'Shanim (Barech Aleinu).

(R. Yehoshua ben Levi): Even though one may request his needs in 'Shome'a Tefilah', he may request at length after Shemoneh Esre, even like the confession of Yom Kipur.

Berachos 34a (Rav Yehudah): One may not request his needs in the first three or last three blessings, only in the middle blessings.

(R. Chanina): In the first three blessings, one is like a slave recounting the praise of his Master. In the middle blessings, he is like a slave requesting remuneration from his Master. In the last three blessings, he is like a slave thanking his Master for giving to him, and he departs.

Rishonim

The Rif (Berachos 22b) bring these Gemaros.

R. Yonah (DH Af): There are four distinctions. 1) One may request his needs in any way in Shome'a Tefilah, for it includes all needs and was enacted for this. 2) One may add at the end of each Berachah requests like the content of the Berachah, even if it is not his personal need, since he requests in the plural. He may not request in the singular, for it is not for all of Yisrael, and it is a Hefsek. He adds in the plural at the end of the Berachah. Since he finished what Chachamim obligated him, he may add. He may not ask in the middle requests for the Rabim, for it looks like he adds to the text that Chachamim enacted. 3) If he requests his own needs, e.g. there is a Choleh (sick person) in his house who needs Refu'ah, he can request in the singular, even in the middle of a Berachah. In Avodah Zarah, we say that he requests for a Choleh in Refa'enu. It did not specify at the end. This connotes that he may request in the middle, since he needs it now and he asks in the singular, so it does not look like he adds to the text that Chachamim enacted. 4) At the end of Tefilah, before or after Yihyu l'Ratzon, he may request in the singular or plural, whether for his own needs or for the Rabim.

Beis Yosef (OC 119 DH v'Chilukim): It seems that R. Yonah learned from Avodah Zarah. It seems that Rav Yehudah (brei d'Rav Shmuel) and R. Chiya teach about the same thing, and so Rashi explains, but R. Yonah explains that Rav Yehudah teaches about adding at the end on behalf of the Rabim, and R. Chiya teaches about requesting one's own needs.

Rambam (Hilchos Tefilah 6:2): One who prays with the Tzibur may not pray too long, but he may say (additions) after his Tefilah, even like the confession of Yom Kipur. Similarly, he may add in each of the middle Berachos similar to its content.

Rambam (3): E.g. if someone was sick, he requests mercy for him in Refa'enu. If he needs income, he requests in Birkas ha'Shanim, and similarly for each Berachah. One may request all his needs in 'Shome'a Tefilah', but not in the first or last three Berachos.

Hagahos Maimoniyos (3): Many permit requesting needs of the Rabim in the first or last three Berachos such as praise and Kavod ha'Yom, which we say in the last three Berachos. This is why we say Krovetz (a Piyut) in the first three Berachos. Ha'Kalir was a Tana, perhaps R. Elazar b'Rebbi Shimon, and he composed a Krovetz. We may rely on him. R. Chananel disapproved. Some forbid saying Zochreinu l'Chayim in Birkas Avos on Rosh Hashanah, for they hold that it is an individual need.

The Rosh (Berachos 5:21) brings the Gemara in Berachos.

Poskim

Shulchan Aruch (OC 119:1): One may add requests in each of the middle blessings. E.g. he requests mercy for a Choleh in Refa'enu. If he needs income, he requests in Birkas ha'Shanim. One may request all his needs in 'Shome'a Tefilah', for it includes all requests.

Kaf ha'Chayim (1): Mahari Molko says that it is better to intend for a Choleh in Refa'enu, and to pray for him in Shome'a Tefilah. Birkei Yosef says that we pray for him in Refa'enu. If one does not want to interrupt his prayer, he intends for him when saying 'Rofei' at the end of the Berachah.

Mishnah Berurah (1): One may request only for present needs in the other Berachos, and in Shome'a Tefilah even for the future.

Mishnah Berurah (4): In Shome'a Tefilah it is good to confess and request income, even if he is rich (Ari Zal and the Zohar). One who sinned should explicitly confess about it in the coming Tefilah, pray with humility and amidst crying, and accept never to willingly do so again. If he does not, this arouses prosecution from above.

Kaf ha'Chayim (5): (Even in Shome'a Tefilah) one should not ask many things. Rather, he makes a general request, e.g. for 'Zera Anashim' (this can mean a male, a Chacham, or to an average child). The Taz (122:2) forbids adding to each Berachah on a fixed basis. This does not include confession and asking for income, which the Ari Zal and Zohar obligate.

Rema: When he adds, he begins the Berachah and later adds, but he may not add and then begin the Berachah.

Taz (1): He must begin with the Berachah to make it primary, and his addition is secondary.

Mishnah Berurah (3): The beginning must be a full matter, e.g. 'You give to people Da'as.' Just 'You' alone does not suffice.

Shulchan Aruch (ibid): R. Yonah says that when he adds in a Berachah like its content, if it is for all of Yisrael, he must request in the plural, and only at the end of the Berachah. If he requests his own needs, e.g. there is a Choleh in his house, or he needs income, he can request even in the middle of a Berachah. , He asks only in the singular. In Shome'a Tefilah, and also at the end of Tefilah, before or after Yihyu l'Ratzon, he may request in the singular or plural, and for his own needs or for the Rabim.

Beis Yosef (DH v'Ha): Our custom is to add in Selach Lanu on fast days, unlike R. Yonah. We can say that a Tzibur is different, just like we say Krovetz in the first three Berachos, even though the Gemara forbids this (i.e. for individuals).

Mishnah Berurah (8): Likewise, the Tzibur says Selichos in the middle Berachah on Yom Kipur.

Beis Yosef (DH veha'Rosh): The Rosh did not mention these distinctions. This implies that he does not distinguish between asking in the middle or the end, or between in the singular or plural. Shome'a Tefilah is like other Berachos, except that in it one may request anything, and in another Berachah one may request only like its content. Rav Yehudah mentioned adding at the end. He does not exclude the middle; he merely discusses a typical case. Alternatively, this is a bigger Chidush, that one need not say what Chachamim enacted just before the closing of the Berachah. All the more so, one may add in the middle. Alternatively, he excludes adding at the beginning. R. Yehoshua ben Levi teaches that after Shemoneh Esre, one may even begin with his additions. R. Chiya bar Ashi held that it is obvious that one must begin with the enacted text of each Berachah, so he did not need to mention 'at the end.'

Kaf ha'Chayim (10): Many rule like the Bach, that even in Shome'a Tefilah one may add only at the end of the Berachah.

Gra (5): We permit requesting at length at the end of Tefilah. It is even more lenient than Shome'a Tefilah. Sa'if 2 forbids requesting at length in the Berachos. Tosfos (8a DH Im) says so.