The period of time that Kaddish is said::[1]

Ideally, one is to say Kaddish for 12 months, however the custom is to only recite Kaddish for a period of 11 months [from the passing[2]].[3] The Avel is to stop reciting Kaddish on the last day of the 11th month; for example if the Yahrzeit is the 10th of Shevat then Kaddish is recited until the 9th of Teves, including the 9th of Teves.[4] [This applies even in a leap year.[5] One is to be very careful in this matter.[6] This applies even if the deceased was known to be a great Tzaddik.[7] Nevertheless, even during the 12th month, participating in the Minyan and answering Amen to Kaddish helps elevate and effect the Neshamah of the Niftar.[8]]

Summary: One begins reciting Kaddish immediately after receiving knowledge of the passing, and concludes the Kaddish 11 months from the passing, on the last day of the 11th month.  For example, if the Yahrzeit is the 10th of Shevat then Kaddish is recited until the 9th of Teves, including the 9th of Teves.   Q&A How does one calculate the end of the eleven months for Kaddish-from the time of death or the time of burial?[9] Some Poskim[10] rule that the eleven months of Kaddish is counted from the day of burial. Other Poskim[11] rule it is counted from the day of the death. Practically, one is to follow the day of death.[12]   How does one calculate the end of the eleven months for Kaddish if one was in a different time zone at the time of death?[13] In the event that the Avel and the deceased were in different time zones at the time of death, the custom regarding Kaddish is to follow the time zone of the deceased with regards to starting the count of the eleven months. If one did not begin saying Kaddish after the burial, for how long is Kaddish to be recited?[14] It is only recited until 11 months from the day of death, even if he only began saying Kaddish on the last day of the 11th month. Is one who is not a child of the deceased [such as a hired person] to recite Kaddish for 11 or 12 months?[15] The custom is to recite it for only 11 months.[16] Nevertheless, some[17] write he may/should recite it for 12 months. Practically, we follow the former opinion.[18] May the Avel recite Kaddish Derabanan in the 12th month? Some Poskim[19] rule he may recite Kaddish Derabanan even in the 12th month. Other Poskim[20] however rule he is not to recite it.

Lechayim:[21] Some are accustomed to make Lechayim in Shul on the day they finish reciting Kaddish.

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[1] Rama 376:4; Maharil Semachos; Levush 133; Beir Heiytiv 132:5 in name of Kneses Yechezkal; See Taz 340:12 and 376:5; Sefer Haminhagim p. 180 [English]; Igros Kodesh 4:106; Sefer Hasichos 1989 1:175 [printed in Shulchan Menachem 5:290-298] Sichas 9^th Shevat 1951; Nitei Gavriel chapter 51

One who knows his father is a Rasha: One who knows his father or mother is a Rasha is obligated to say Kaddish for 12 months. [Chomos Yerushalayim 257, brought in Pischeiy Teshuvah 376:3; See Chasam Sofer E.H. 1:69]

One who committed suicide: If the deceased committed suicide it is permitted to say Kaddish for 12 months on his behalf. [Chasam Sofer E.H. 1:69]

One whose father told him to say Kaddish for 12 months: If one’s parent commanded him to say Kaddish for 12 months then he is to do so. [See Beis Yitzchak 2:157; Mishmeres Shalom Tzaddik 26 that so did the son of the Maharshak for his father; Shevet Halevi 2:161 and 165; Igros Kodesh 4:140 regarding Reb Nissan Tloshkin who asked for Kaddish to be recited for 12 months; Poskim in Pnei Baruch 34 footnote 4 and Nitei Gavriel 51:10]

One who does not know the date of his father’s passing: See Pnei Baruch 34:6

If another relative passed away within the 11 months, may he say Kaddish in the 12^th month of the parent? See Sefarim in Igros Kodesh 12:197; The Rebbe there concludes it may be recited; See also Gesher Hachaim 30:9-13; Chelkas Yaakov 3:157; Pnei Baruch 34:7

Other customs: Some Poskim rule one is to recite Kaddish for 12 months minus one week. Meaning they stop saying Kaddish on the last week of the 12^th month. [Birkeiy Yosef 376 based on Arizal that the Kaddish helps even souls that are not in Gehinnom; See Rav Poalim 3:32; Poskim in Pnei Baruch 34 footnote 4; Various opinions brought in Igros Kodesh 4:106 footnote 1]

[2] See Q&A!

[3] The reason: This is done in order not to turn one’s father and mother into Reshaim, as the judgment of a Rasha is 12 months. [Rama ibid] The mere ending of the Kaddish after 11 months has the ability to refine the judgment of the deceased and cause him to be elevated above. [Sefer Hasichos 1989 1:175, printed in Shulchan Menachem 5:290]

[4] Beir Heiytiv 132:5; Shvus Yaakov 2:129; Machatzis Hashekel on M”A 132:2; Chayeh Adam 32:18; Kitzur SHU”A 26:17; Sefer Haminhagim p. 180 [English]; Igros Kodesh 4:106; Sefer Hasichos 1989 1:175 [see there for explanation]; Igros Kodesh Rashab Halacha 80; Vol. 2:904; [See Sichas 9^th Shevat 1951 that the Rebbe Rashab once added one day to the Kaddish and he was very distressed]; See Pnei Baruch 34 footnote 3 for other Poskim and opinions

If the last day is Friday: Some Poskim rule that if the last day is Friday one is to say Kaddish also on Shabbos. [See Poskim in Pnei Baruch 34:8; See Shevet Halevi 165 that he is not to say Kaddish]

Other customs: Some are accustomed to end the Kaddish 30 days before the 12^th month. [P”M, in end of Noam Megadim, and so was the custom of the Rebbe Rayatz, brought in Sefer Hasichos ibid]

[5] Poskim brought in Nitei Gavriel 51:3

[6] Sichas 9^th Shevat 1951 that the Rebbe Rashab once added one day to the Kaddish and he was very distressed

[7] Sichas 9^th Shevat 1951 that so was the custom of the Rebbe Rashab, and so followed the Rebbe, see there for the explanation; Sefer Hasichos 1989 1:175

[8] See Sefer Hasichos 1989 p. 182

[9] See Igros Kodesh 3:199; 4:107 [published in Shulchan Menachem 5:292-295]; Gesher Hachaim 30:9-10; Nitei Gavriel 51:4 footnote 6-8; Pnei Baruch 34:9

Background:

There is known dispute regarding whether the 12 months of Aveilus is counted from the day of death/burial, and whether the Yahrzeit is the day of death or burial, and whether the 12 months in Gehinnom is counted from the day of death or burial. It is possible to learn that regarding Kaddish there is a similar dispute, as depending on when the 12 months of Gehinnom concludes would depend on when to conclude Kaddish [see Kneses Yechezkal 47]; On the other hand perhaps it depends on the start date of the Kaddish, whether one started the day of burial or the day of death [see Daas Torah 376:6].

[10] Implication of Shach 402 in Nekudos Hakesef that Mishpat Reshaim of 12 months begins from day of burial [see Kneses Yechezkal 47]; Levush brought in Elya Raba 132:4; Noam Megadim 1; Kitzur SHU”A 26:17; Neharei Afarsimon 97; Biur Halacha 132; See Daas Kedoshim 377; Levushei Mordechai 223

[11] Implication of Taz 568:4 that Mishpat Reshaim of 12 months begins from day of death; Beir Heiytiv 132:5; Siddur Yaavetz 39 [see Nitei Gavriel ibid footnote 6]; Kneses Yechezkal 47; Mateh Ephraim 4:1; Alef Lamateh Kaddish Yasom 3:5

[12] Igros Kodesh 4:107; Custom of Rebbe Rayatz as brought in Reshimos Hayoman p. 416; However, see Igros Kodesh 3:199 that the Rebbe directed the asker to follow the day of burial and not the day of death! See also Igros Kodesh 13:410 in which the Rebbe implies to follow the day of the burial. Vetzaruch Iyun!

The reason: As Admur 71:1 rules to begin saying Kaddish the day of death and according to some Poskim [see Daas Torah ibid] the dispute is dependent on this matter.

[13] Igros Kodesh 13:410, brought in Shulchan Menachem 5:295

[14] See Taz 568:4; Shach 402 in Nekudos Hakesef; P”M 132 A”A 2; M”A 132:2 in name of Ran; Beir Heiytiv 132:2; Rav Akiva Eiger 376 and Beis Lechem Yehuda 376; Gesher Hachaim 30:9-5; Sefer Haminhagim p. 180 [English]; See Pnei Baruch 34:4-5

[15] See Pnei Baruch 34:11

[16] See Shevet Halevi 165 that if the son hired him then he may not say it in the 12th month; Gesher Hachaim 30:9-4 that so is custom of some; See Pnei Baruch ibid; Nitei Gavriel 50:15; The Rebbe recited Kaddish for only 11 months for the Rebbe Rayatz and the Rebbetzin; See Sichas 9^th Shevat 1951 and Sefer Hasichos 1989 of 21 Teves that the Rebbe stopped saying Kaddish after 11 months and explained why. The Rebbe was not a son or daughter!; Rav Y.L. Groner in a written correspondence that the Rebbe told people to say Kaddish for his brother and for the Rashag only for 11 months; Rav Y.S. Ginsberg that so is the custom;

[17] Gesher Hachaim 30:9-4; Meishiv Devarim 213; Nitei Gavriel 50:19

[18] Nitei Gavriel 50:15; See sources in previous footnote that so is the custom and so was the Rebbe’s custom

[19] Rav Poalim 4:32; Gesher Hachaim 30:9-10; Shevet Halevi 165; See Even Shoham 20, brought in Pischeiy Teshuvah 376:4; See Pnei Baruch 34:10

[20] Maharshag 52; Even Yaakov 49; See Pnei Baruch ibid

[21] Nitei Gavriel 52:4 in name of Rav Poalim 3:62 [I did not find this in the source mentioned in Rav Poalim]