First of all, if you don't like reading scripts, you are not going to like this. This is the script of the play, though I wonder how they staged some of it. But that's the magic of theater, isn't it?
The story picks up right where Book 7 (Harry Potter and the Deathly Hallows) leaves off, at the Epilogue in Kings Crossing. It even replays that scene.
We learn that it isn't easy being the son of Harry Potter (and it still isn't easy being Harry Potter), nor being the son of Draco Malfoy. Albus Severus Potter meets Scorpius Malfoy, and they become friends, which is facilitated by Sorting Hat placing Potter in Slytherin. (Gasp!) Who is this Cursed Child of the title? Is it poor Scorpius, rumored to be the son of Voldemort and Astoria Greengrass, Draco's late wife, who was sent back in time to be impregnated, to carry on the Malfoy name when Draco wasn't up to the task? (Such an idea seems silly on the face on it, but I think it served one purpose: to plant the seed that some Time Turners still exist).
(Here's where I wish I had written this up back in January -- the point of this blog is to help me remember these details!)
As the story unravels, we meet Amos Diggory, father of the deceased Cedric, and Amos's niece. The three of them hatch a plan to steal a Time Turner (there is a secret one, which survives, should it prove needed) and go back and save Cedric's life. But whatever they do, playing with time, has serious consequences, including people disappearing (such as Ron and Hermione's daughter, Rose, because the two never got together).
As you can imagine, they have to fix things, without making things worse, including confessing to the people in a doomed world what's going on, to get their help to set things right.
I can agree with someone's online assessment that it's a nice story, but I wouldn't want it to be canon. It reads like fan fiction, but only because it's a little light in its words -- it is a script, after all. Plus, there are limits to theater.
Unlike each Potter book, this one spans several years, and there are big gaps, where you might wonder what the characters are doing. It tells a story, but it closed off others and will make life difficult for other stories to take place. I guess we all have ideas where we'd like the Potter people to go and to do afterward -- or even in those missing years before the Epilogue.
Statistics: library e-book loan, script, continued series