Review written in 2016:
I went into this book with little to no expectations and honestly struggled to find bits in it to enjoy. My favorite part was the new characters- Scorpius and Albus. I was glad that both of them were different than their fathers, and enjoyed the arc of their friendship. There are some genuinely funny moments of dialog between the two of them, and Scorpius's obvious massive crush on his friend tickled my slash-driven heart.

HOWEVER. (There will be some spoilers following this. This is the warning.)
The whole plot of the book is RIDICULOUS.

The plot heavily revolves around time-travel, which is not a trope I generally enjoy. JKR did a good job with the limited scope of the Time Turner in book 3, but in this play the rules governing time travel are inexplicably completely different. Early on, in a very heavy handed scene, we are introduced to a rumor that Scorpius's mother was sent back in time via a Time Turner and that he is Voldemort's son. I HAVE SO MANY PROBLEMS WITH THAT, the main one being rape but the secondary one being that I don't think Voldemort cared about/would be capable of sex. He had no interest in having an heir, because he planned to live forever. Why would someone who wants to be eternal need a legacy? A bit later on, Albus witnesses Amos Diggory berating Harry Potter for the death of his son. For some reason this causes Albus to decide that he and Scorpius should break into the Ministry, steal a Time Turner and save Cedric Diggory. Why? Because Albus is an angsty teen burdened by this father's fame. Naturally.

Albus and Scorpius first travel one, then two layers away from their own time steam, fucking up the world more with each progressive change. It's like Back to the Future, but less charming. The only reason for this that I can fathom from a writings standpoint is that it allows our young heroes a chance to interact with characters who are dead. I will say that the scene between Snape and Scorpius was actually one of my favorite scenes, but it wasn't good enough to justify the huge amount of bullshit it took to get there.

The book also gets some really basic things wrong about magic usage in the Harry Potter universe. At one point a character offers to mix up a batch of Polyjuice Potion to be used immediately, but we all know that that potion takes a month to prepare. It's a small mistake, but a good example of why this book just doesn't FEEL like a real Harry Potter story. I've read fan fiction that did better.

Edited added in 2020: as of this year, I no longer plan to read any new book that JKR is involved with.