Gemara

(Mishnah): He holds Shtei ha'Lechem and the lambs, and is Molich u'Mevi (moves them outward and brings them back), and is Ma'aleh u'Morid (lifts them and brings them down).

62a (R. Chiya bar Aba): He is Molich u'Mevi to the Owner of the four directions. He is Ma'aleh u'Morid to the Owner of Heaven and earth;

(Rav Chama bar Ukva): He is Molich u'Mevi (to ask Hash-m) to withhold bad winds. He is Ma'aleh u'Morid to withhold bad dew.

(R. Yosi bar Rav Avin): This teaches that even a Mitzvah (Tenufah) that is not Me'akev withholds punishments (bad winds and bad dew).

(Rabah): Also Lulav is shaken (with the other three species) in this way.

Rav Acha bar Yakov did so and said 'Satan, this is an arrow in your eye!'

This was wrong. One should not taunt the Satan, lest this incite the Satan against him.

Sukah 37b (Mishnah - Beis Hillel): One shakes in Hodu la'Shem Ki Tov, at the beginning and the end, and at Ana Hash-m Hoshi'ah Na.

Berachos 30a (Beraisa): If one must travel early, he shakes a Lulav before leaving. When the time for Shma comes, he says Shma;

Nidah 62a (Mishnah): We pass seven ingredients over a stain to determine if it is Dam Nidah. Each ingredient must be rubbed three times (on the stain).

63a - Question (R. Yirmeyah): Does rubbing back and forth count like once, or like twice?

This question is not resolved.

Rishonim

Rif (Sukah 18b): Some say that one is Molich u'Mevi three times in addition to Molich u'Mevi and Ma'aleh u'Morid (mentioned in the Mishnah). They learn from the Yerushalmi. It says that one must shake three times on everything. R. Zeira asked whether both, i.e. Molich u'Mevi, count like two times, or like once. This shows that the three times are in addition to Molich u'Mevi and Ma'aleh u'Morid. (I.e. if 'three times' included Molich u'Mevi and Ma'aleh u'Morid, R. Zeira would know that Molich u'Mevi cannot count like two, for if so, the same applies to Ma'aleh u'Morid, and there are four in all! - PF)

Ran (DH Lenane'a): The primary Mitzvah is shaking it. If one must travel early, he shakes a Lulav before leaving. Rav Acha bar Yakov shook it this way and said 'Satan, this is an arrow in your eye!' Surely, this was at the time of Netilah (after blessing on it and being Yotzei the Mitzvah). He would not interrupt Hallel to say this!

Ran (DH Molich): One is Molich u'Mevi (Shtei ha'Lechem and the lambs, in two directions), for the One Who rules over two directions surely rules over all four! He intends for the One Who rules over all four directions. This is the custom (regarding Lulav).

Ran (DH Amar, form the Ritz Gei'us): It suffices to be Molich u'Mevi and Ma'aleh u'Morid. The simple reading of Rabah supports this. They must explain the Yerushalmi to say that whenever he shakes, i.e. at the time of Netilah and every place in Hallel that we shake it, he shakes three times. R. Zeira did not ask about Ma'aleh u'Morid. One cannot leave it up. He must lower it again, so this counts like one. He asked about Molich u'Mevi. Perhaps one must be Molich u'Mevi, Ma'aleh u'Morid, and again be Molich u'Mevi. Or, even though Ma'aleh u'Morid counts like one, Molich u'Mevi counts like two, so it suffices to be Molich u'Mevi and Ma'aleh u'Morid. The question was not settled, and the Bavli did not say that he is Molich u'Mevi twice. This connotes that Molich u'Mevi counts like two, so it suffices to be Molich u'Mevi and Ma'aleh u'Morid.

Rebuttal (Ran): If so, why do we require that the Lulav be a Tefach above the myrtle, to enable shaking it (Sukah 32b)? One can be Molich u'Mevi a Lulav of any size! Rather, one must shake the Lulav and rustle the leaves. The Rambam says so. One shakes the Lulav three times during Holachah, and similarly when he is Mevi, Ma'aleh and Morid. He explains the Yerushalmi to say that one must shake (three times) during every movement of Molich u'Mevi and Ma'aleh u'Morid. Then R. Zeira asked whether Molich u'Mevi count like one, and one shakes three times in all during Molich u'Mevi, and the same applies to Ma'aleh u'Morid. Or, Molich u'Mevi are separate, and one shakes three times during each by itself. The question was not resolved. Therefore we are stringent, and we shake three times during each. The Rif holds like this.

Ran (ibid.): Others explain that one must shake three times on (aside from) the places in Hallel and at the time of Netilah. We asked if Molich u'Mevi count like one, i.e. one must be Molich u'Mevi (a small amount) three times, or if Molich u'Mevi counts like two, and it suffices to be Molich u'Mevi u'Molich. The Yerushalmi supports this. It compared this to R. Yirmeyah's question, if rubbing back and forth counts like once, or like twice. This is exactly like R. Zeira's question. The question was not resolved, so we are stringent. One is Molich u'Mevi, and Ma'aleh u'Morid, and after each of these (four) he shakes the Lulav three pairs of small movements (back and forth). Also the Ramban says so.

Rambam (Hilchos Lulav 7:10): One is Molich the top of the Lulav three times, and is Mevi and shakes the top three times. The same applies to Ma'aleh u'Morid.

Hagahos Maimoniyos (50): The Yerushalmi did not resolve whether going back and forth counts like twice or like once. We are stringent.

Hagahos Maimoniyos (60): Avi ha'Ezri says that the shakings are at the time of the Berachah, but at the beginning and end of Hodu one merely shakes it (a little). R. Simchah says that the Gemara discusses shaking during Hallel, but there is no source to shake after Netilah. The only source is one who must travel early; he shakes a Lulav before leaving. Perhaps there is different, for he will not have a Lulav at the time of Hallel! The custom is to shake at the time of Berachah and during Hallel. It is astounding that the Mishnah about where to shake does not mention after Netilah.

Rosh (Sukah 3:26): We do not find that one must shake the Lulav at the time of Netilah, but we obligate a child who knows how to shake (due to Chinuch - 42a). This implies that he shakes at the beginning of Netilah, even if he is too young to know to recite Hallel. Also, if one must travel early, he shakes a Lulav before leaving. Avi ha'Ezri brings a proof from the Yerushalmi. It says that we shake in order to shake (thwart) the power of the Satan. Rav Acha bar Yakov brought it and shook it this way at the time of Berachah, and taunted the Satan. Avi ha'Ezri says that the primary shaking is at the time of the Berachah. Shaking during Hallel is mere shaking; about it, the Yerushalmi says that it is to shake the Satan's power. It then says that Rav Acha bar Yakov brought (e.g. Molich or Mevi) it and shook it this way at the time of Berachah; the previous shaking mentioned (to shake the Satan's power) was just shaking, without bringing it. This is like R. Yitzchak taught, that we blow the Shofar, and blow again, to confuse the Satan. The first blowing is primary; the second is to confuse the Satan. The same applies to shaking.

Rosh (DH Gemara): Some say that one shakes three times while being Molich u'Mevi, aside from Molich u'Mevi and Ma'aleh u'Morid. They learn from the Yerushalmi. 'On every matter' means in every direction that he is Molich u'Mevi, he must shake three times, aside from Molich u'Mevi. I.e. Molich u'Mevi is not enough. He must shake three times in each Holachah and Hova'ah. We would explain that he holds his hand in place and merely shakes, if not for R. Zeira's question, which implies that he must be Molich u'Mevi three times in every direction; he asked whether Molich u'Mevi count like one or like two. Tosfos, and R. Chananel in the name of Rav Hai Gaon, say that we are stringent, for the question was not resolved. Even though shaking is mid'Rabanan, we are stringent due to dearness of the Mitzvah. One extends three times, and brings it back three times, in each of the six directions (including up and down), 36 shakings in all.

Rosh (citing Ba'al ha'Itur): Since it is mid'Rabanan, we are lenient. Our custom is to just be Molich u'Mevi, and Ma'aleh u'Morid. The One Who rules over two directions rules over all four. It is an outsider's opinion to be Molich to north and south. (I.e. one faces east in the Beis ha'Keneses, so it suffices to be Molich u'Mevi forwards and back, i.e. east-west - PF.)

Rebuttal (Rosh): The custom is to be Molich in all four directions; this is proper. We cannot praise enough the One Who rules over all four directions! The other custom is of outsiders. The one who initiated it was concerned lest shaking in all four directions resemble a cross. Just the contrary! If one shakes in just two directions, and also up and down, this is like a cross, for there are four extremities! If one shakes in four directions, and also up and down, there are six extremities.

Poskim

Shulchan Aruch (OC 651:9): To shake, one is Molich and shakes three times b'Holachah, and three times in Hova'ah.

Taz (10): The Rosh cannot be understood simply. If not for R. Zeira, we would say that he holds his hand in place and merely shakes. If so, we can say that he shakes three times through Molich u'Mevi after each Holachah and after each Hova'ah, but he is Molich u'Mevi only once in every direction! (Kitzur) Piskei ha'Rosh says so.

Note: Kitzur Piskei ha'Rosh says 'he is Molich and shakes there b'Holachah and Hova'ah and brings it to himself and he does three times. Similarly, to north and south he does so three times and is Molich and does so [also above and below], in all 36 times.'

Taz (10): The Beis Yosef fixed the Tur's text to be like Kitzur Piskei ha'Rosh. This is like the third opinion in the Ran. Even though the Ran connotes that only three small pairs (shakings back and forth) are needed in every direction, and the Rosh requires six, they agree that the shakings are after Holachah and Hova'ah. However, if the shakings are unrelated to Molich u'Mevi, what was R. Zeira's question? Rather, one shakes during Holachah and Hova'ah. We are stringent to require three Holachos and three Hava'os in every direction.

Mishnah Berurah (42): This opinion holds that he is Molich u'Mevi once in each direction, and shakes three times during Holachah and three times during Hova'ah.

Kaf ha'Chayim (93): The proper custom is what everyone does, i.e. to be Molich u'Mevi three times in every direction, like the Ari Zal says.

Kaf ha'Chayim (48): During Holachah and Hova'ah, one must keep the Lulav next to the Esrog. One brings them back even with his chest. Some move the Lulav a little at a time, in order to make three shakings (motions) in one Holachah, and three in one Hova'ah. This is wrong.

Rema: He shakes the Lulav and rustles the leaves in each shaking.

Mishnah Berurah (43): He rustles a little. One who rustles so much that he almost breaks the Lulav, he errs.

Shulchan Aruch (ibid.): Afterwards, he tilts his hand to another direction and does so. He does so in each of the four directions, and above and below.

Rema: The Molich u'Mevi itself is the shaking. He does so three times in every direction.

Mishnah Berurah (44): He holds that one must be Molich u'Mevi three times in every direction. He shakes and rustles the leaves during the Holachah and Hova'ah.

Gra (DH veha'Holachah): In Darchei Moshe, the Rema said that this is from R. Avraham of Prague. It is unlike the Shulchan Aruch, who says that one shakes three times after each Holachah and after each Hova'ah.

Magen Avraham (20): The Maharshal says that to be stringent, one is Molich u'Mevi three times (in every direction), and shakes three times during Holachah and three times during Hova'ah. One who does more is called a commoner. It seems that he holds that he shakes three times where Holachah or Hova'ah finishes. The Bach says that it suffices to shake during Holachah and Hova'ah. This is the custom.

Mishnah Berurah (46): (When shaking down,) it is best not to invert the Lulav, like the Taz says.

Kaf ha'Chayim (99,100): The Ari Zal says that it should be held vertical for all the shakings. Also, it is disgraceful to hold it upside down.