Harry Potter, the Boy Who Lived and Ministry of Magic bureaucrat

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Harry has always been a famous wizard, and in the 7th book, he has turned into a living legend when he defeated Voldemort.
During the present timeline of Cursed Child, he's the overworked Head of the Department of Magical Law Enforcement who prefers action to paperwork. Although he's Master of Death by the end of Deathly Hallows, he hasn't quite mastered parenting, and he has to confront the ghosts of his past.

Photo by Manuel Harlan, courtesy of Pottermore
Out of Hogwarts, Ron, Hermione, and the rest learn "How to Adult"

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Ron and Hermione are the heads of the Granger-Weasley household, with their daughter, Rose, who is as ambitious as her mother and is a gifted Quidditch Chaser like her Aunt Ginny.
Hermione has risen to the rank of Minister of Magic, while Ron still loves to send joke items from Weasleys' Wizard Wheezes, which he now helps his brother George run.
Other characters have taken on good jobs. Ginny, Harry's wife, edits the sports pages of the Daily Prophet, while Neville Longbottom is professor of Herbology at Hogwarts.
Albus Severus Potter, "Slytherin Squib"

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As his father, Harry Potter, is an icon of the wizarding world – having defeated the most dangerous dark wizard of all time – Albus carries a heavy burden on his shoulders.
Albus is quite the angsty one among his siblings, and perceives himself as a disappointment to the family name, and there's a steady tension between father and son.
He's not exactly what you'd expect a Potter to be: he gets sorted in Slytherin, is not exactly gifted with spells, and hates Quidditch – in spite of his father being a star Seeker. At one point, he even gets teased as a "Slytherin Squib." It doesn't help either that he's told something along the lines of "Why can't you be more like your siblings James Sirius and Lily Luna?"
Albus Potter and Scorpius Malfoy, BFFs

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You would also expect Albus to hang around with the kids of Harry's best friends, Ron and Hermione, but on their first ride aboard the Hogwarts Express, he makes an unlikely friend: the son of his father's former nemesis Draco Malfoy, Scorpius.
The two are inseparable, but like Albus, Scorpius is not exactly who you would expect a Malfoy to be.
Rose Granger-Weasley: Not quite Albus' friend
Among his relatives, we see Rose a lot more in Cursed Child. She warns Albus about making friends with Scorpius, and when Albus ignores this, things between them turn sour.
The Malfoys are not evil anymore.

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The Malfoys played a crucial part in the fall of Voldemort – after all, it was Narcissa Malfoy who had lied about Harry's death and Draco was at one point, the owner of the Elder Wand.
Now, it's just Draco and his son, Scorpius, a self-proclaimed geek reminiscent of Hermione, but according to his father, "more of a follower than a leader" (which makes an interesting dynamic between him and the headstrong Albus).
However, Draco was a former Death Eater, and very malicious rumors surround the family. Some people say that Scorpius is secretly the son of Voldemort.
Amos Diggory wants justice – and his son back
Amos Diggory, a bereaved father, wants his son back.
We remember Cedric Diggory being killed as the "spare" when the Triwizard Cup-portkey sent him and Harry straight to Voldemort. Amos sees this as a reason that his son could be brought back to life.
But how is this possible if the Killing Curse ("Avada Kedavra") is irreversible?
Rewriting history with a Time-Turner
Time-Turners are contraband magical items in the storyline of Cursed Child. But one emerges, and by his fourth year at Hogwarts, Albus attempts to do what his father couldn't in Goblet of Fire: save Cedric Diggory.
However, changing events creates alternate timelines – and this includes repercussions in one key event in wizarding history: the Battle of Hogwarts.
Iconic characters return
Through flashbacks and whatever magical means possible, characters from the past – beloved and hated – are seen again in Cursed Child.
In alternate timelines created by tampering with history via a Time-Turner, some people are well and free (e.g. Dolores Umbridge) – and alive: Severus Snape.
The wise and late Hogwarts Headmaster Albus Dumbledore makes an appearance, too, via his portrait, that is.

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Other returning characters include Moaning Myrtle, Hagrid, Professor McGonagall, and Bane the centaur.
Harry's scar hurts again

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We don't want to reveal too much, but Harry's scar hurts again, and the implications of that are tackled in the book in the most intense (and awesome!) way.
When was the last time this happened? Obviously, when Voldemort was still around. *cue suspense music*

Photo by Manuel Harlan, courtesy of Pottermore