One of the aspects of Harry Potter fan culture that's not discussed as much is the "next generation", aka the period of time after the main series that follows the children of the Harry Potter characters. A lot of this was just based on fan theories and occasional postings/updates J.K. Rowling would put on Pottermore, the official Harry Potter fan website. When The Cursed Child was released, suddenly these kids were all a bigger part of the canon, and fans got a lot more information on these characters that had only been tiny parts of the plot for so long.
But even The Cursed Child doesn't totally cover all of the children that the main trio of characters from Harry Potter had, as it only focuses on a couple of them. Anyone who's only familiar with the main Potter stories also might not be super familiar with the kids that Harry, Ron, and Hermione had, as they're only mentioned briefly in the epilogue of the last book. How many kids did each of them have, and what pieces of information are known about them?
How Many Kids Did Harry and Ginny Have?
A big plot point in the novels is Harry and Ginny's romance, and it apparently worked out for them as they stayed together, got married, and had kids in the future. The Cursed Child closely follows one of those kids, Albus, named after Dumbledore because of his and Harry's close connection. Albus is also the kid that gets the most focus in the epilogue of The Deathly Hallows, as he discusses his fear of being sorted into Slytherin with Harry. Besides Albus, who is the middle child, Harry and Ginny also had two other kids named James and Lily, after Harry's parents.
James was the oldest of the three and was sorted into Gryffindor at Hogwarts. He was named not only after Harry's father, but also his godfather, as his middle name is Sirius. Lily was the youngest of the Potter children and was also a Gryffindor. Her middle name was Luna, as an ode to Luna Lovegood who was close with Harry and Ginny. Harry and Ginny had these three biological children, but they also spent a lot of time with Teddy Lupin, the son of Remus Lupin and Nymphadora Tonks, as Harry was his godfather. Both of Teddy's parents died during the Battle of Hogwarts, so he was raised by his grandmother and spent a lot of his time with the Potters and Weasleys. He was sorted into Hufflepuff and was a Metamorphmagus like his mother, and is probably most known to Harry Potter fans for dating Victoire Weasley, Bill and Fleur Weasley's oldest daughter.
How Many Kids Did Ron and Hermione Have?
As most fans will remember, Ron Weasley and Hermione Granger got together at the end of the series, and ended up getting married and having kids just like Harry and Ginny did. Ron and Hermione only had two children, one of whom also played a big role in The Cursed Child. Rose was the oldest, and she was close with her cousin Albus as they were the same age. The sorting hat placed her in Gryffindor, and she and Albus ended up drifting apart because she disapproved of his friendship with Scorpius Malfoy (unbeknownst to Rose, Scorpius actually had quite a big crush on her). Hugo was the second child, and was a few years younger than his sister. There's not much more information on him, as he doesn't appear in The Cursed Child or the Pottermore posts very often.
Who Are Some of the Other Children of Harry Potter Characters?
There are a few other notable characters from the series that have children that went to Hogwarts around the same time as the children of the Golden Trio. Scorpius Malfoy is probably one of the most well-known, as he plays a big role in The Cursed Child. He's the son of Draco Malfoy and his wife Astoria, and he's best friends with Albus Potter. The aforementioned Victoire Weasley is also one of the more well-known next generation characters, partially due to her relationship with Teddy being the subject of some Pottermore posts. Victoire had siblings as well, as Bill and Fleur also had another daughter named Dominique and a son named Louis. All of these children were part Veela, inheriting those traits from their mother.
There are also some characters who don't have a lot of information out there other than their names and parentage. George Weasley and Angelina Johnson got married and had two children - Roxanne and Fred (named after George's twin who died in the Battle of Hogwarts, of course). Again, there is unfortunately not much information on these two. Luna Lovegood also had twin boys with her husband Rolf Scamander (the grandson of Newt Scamander of Fantastic Beasts and Where To Find Them fame), named Lorcan and Lysander. They don't show up in the books at all, but their existence has been confirmed by J.K. Rowling, though nothing else is really known about them.