How many children must one have to fulfill the Mitzvah of having children?
Biblically: When a man fathers a boy and a girl, he fulfills the Mitzvah of Peru Urevu [as soon as the child is born]. [If he only fathered one gender, either male or female, then he does not fulfill his obligation even if he has many grandchildren of both genders.] Nonetheless, certain criteria must be met for these children to be valid for the Mitzvah of Peru Urevu.
- The children can procreate: In order for a child that is born to be valid towards one’s fulfillment of Peru Urevu, the child must be physically capable of bearing children. This applies for both the son and daughter. Thus, if the son is a Saris [impotent] or the daughter is an Aylanis, they are invalid in counting towards one’s fulfillment of the Mitzvah of Peru Urevu. [It however is not necessary for the son and daughter to actually have children, in order to be valid for their father’s Mitzvah of Peru Urevu. Thus, even if the son or daughter marries a spouse who cannot have children, or does not get married at all, the father fulfills his obligation.]
- The children do not pass away in the father’s lifetime: If both the lone son and daughter passed away [in the lifetime of the father], then the father only fulfills the mitzvah of Peru Urevu if his deceased children left him a male and female grandchild, having at least one child of different gender from each the son and daughter. Thus, whether the daughter left him a son or daughter, and whether the son left him a daughter or son, he fulfills his obligation so long as he has a son from one child and a daughter from another child. It does not suffice if only one child left him children, even if that child left him a son and daughter, if the other child did not have any children. It likewise does not suffice if a son and daughter left him children of the same gender, such as two sons or two daughters. [However some Poskim are lenient and rule that it suffices if he has two male grandchildren, one from each child. Furthermore, some Poskim rule that even having female grandchildren suffices, if he has one from each child. According to the former approach, if one’s only son or daughter passed away then the child who passed away is not considered as part of the necessary two children to fulfill the Mitzvah of Peru Urevu, unless that child left a child of their gender, and one in total has descendants who are male and female. Thus, for example, if the son passed away and left a son as a child, then that Grandson together with his daughter are considered part of his fulfillment of the mitzvah. Likewise, if the daughter passed away and left the daughter as a child, then the son and granddaughter together help him fulfill his mitzvah. However, if the lone son who passed away left only daughters, or the lone daughter who passed away left only sons, then he does not fulfill the Mitzvah if he only has descendants of one gender.]
- The mental state of the child: A child who was born a Mamzer [to the knowledge of the public], Shoteh, or Cheresh, is valid to count towards the obligation. Certainly, a child who is still below Bar/Bas Mitzvah counts towards the obligation.
- A Jewish born child: [If one fathered a child with a gentile woman, he does not fulfill his obligation.] However, if a gentile converted together with his family, then if he has a son and daughter who converted, he fulfills his obligation. [Furthermore, some Poskim rule that he fulfills his obligation even if they do not convert.]
- The child is born from intercourse: Some Poskim question whether a child that was fathered without intercourse [i.e. Nisabra Bambatya; IUI; IVF; Sperm donor] counts as part of one’s obligation of Peru Urevu. Other Poskim conclude that it is valid.
The Rabbinical Mitzvah of Liareiv: All the above is only in regards to the Biblical Mitzvah of Peru Urevu, however, Rabbinically, even if one already has children and fulfilled the Mitzvah of Peru Urevu, he Rabbinically remains obligated in the Mitzvah of Liareiv, to further procreate the world.
 Michaber E.H. 1:5; Beis Hillel in Mishneh Yevamos 61b
Other opinions: Some hold that in order to fulfill the Mitzvah of Peru Urevu one must father two sons. [Beis Shamaiy ibid]
 See Rama 1:6 that the Mitzvah is fulfilled even with a Katan
 Beis Shmuel 1:10
 Michaber 1:5; Braisa Yevamos ibid
 An Aylanis comes from the term Ayal, which means a male ram. A woman who has the nature of a male, and thus does not grow breasts , and her voice is thick like a male, and her vaginal area does not protrude like other woman, is considered an Aylanis. [Pirush ibid]
 Chelkas Mechokeik 1:6; Beis Shmuel 1:8
Other opinions: Some Poskim question whether the father fulfills his obligation if the son and daughter do not have children, such as they did not get married or married a spouse who cannot have children. [Bnei Ahuva 15:1; Pischeiy Teshuvah 1:8]
 Michaber 1:6; Rambam Ishus 15:5
 Pischeiy Teshuvah 1:8 based on Chelkas Mechokeik 1:6 and Beis Shmuel 1:8 who rule it is not necessary for them to have children to validate the father’s Mitzvah.
 Implication of Michaber ibid and Rambam ibid; Beis Shmuel 1:9; Tur Y.D. 392; Semag Asei 49; Semak 284; Perisha 1:17; Bach 1:5
 Tosafus Yevamos 62b, brought in Beis Shmuel ibid and Chelkas Mechokeik 1:7
 Our Girsa in Tur 1, brought in Beis Shmuel ibid; Shiltei Giborim Yevamos 19b; Chelkas Mechokeik 1:7
 Beis Shmuel 1:10; Taz 1:8; However, see Chelkas Mechokeik 1:7 who is in doubt in this matter
 Rama 1:6
 See Sefer Chassidim 500; Beis Shmuel 1:11
 Michaber 1:7; Rambam Ishus
 Chelkas Mechokeik 1:8 in name of Maharil 223; See Beis Shmuel 1:12
 See Mishneh Limelech Ishus 15:4; Pischeiy Teshuvah 1:9
 Chelkas Mechokeik 1:8; Taz 1:8
 Beis Shmuel 1:10 based on Bach Y.D. 195
 See Michaber E.H. 1:8; Beis Shmuel 1:1 and 14-15; Taz E.H. 1:1; Rambam Ishus 15:16; Yevamos 62b; See Likkutei Sichos 30:264 for a dispute if the Mitzvah is due to Liareiv or is a Rabbinical command of Peru Urevu and that according to the Rambam ibid there exists a Rabbinical obligation of Peru Urevu, and that one fulfills the Biblical Mitzvah of Peru Urevu by every additional child.