Harry Potter and the Cursed Child ain't Shakespeare—no one is wearing a weird neck-doily or speaking in sonnets—but it has many hallmarks of great Shakespearean drama.
It features a huge cast of characters. It tackles historical events (try and tell us the Battle of Hogwarts didn't actually happen). It features some slapstick comedy, a humorous case of mistaken identity, and duels…even if these duels are with wands instead of swords.
It's also a tragedy, of sorts, one about the death of a king. Sure, Harry Potter isn't actually royalty (and he doesn't die), but he's pretty close to royalty in the wizarding world, and this story might be the one that closes the book on his adventures.