The Taming of the Shrew
The Taming of the Shrew is most definitely a "Comedy" – a generic category that has a few basic rules and operating principles: a light and humorous tone; clever language and witty banter; deception, disguises, and cases of mistaken identity; young love that must overcome some kind of struggle; family drama; lots of twists and turns; and multiple plots that come together in the end – which always involves the reunification of the family and/or a marriage.
Critics have pointed out, too, that Shakespeare borrows some from the Italian tradition of Commedia dell'arte, a style of improvisational theater involving stock characters (like the Pantaloon, a ridiculous old man trying to get with a much younger woman – think of Gremio) and ridiculous slap-stick humor (like the time Grumio and Petruchio's miscommunication about knocking on Hortensio's door results in Grumio getting beat down by Petruchio).
That said, there are some very dark undertones in the play. For instance, even as the audience laughs at the back and forth play between Kate and Petruchio, they're made uncomfortably aware that Petruchio's behavior is, well, abusive. His taming techniques also involve torture and there are several references to Kate's death throughout the play.
What are we to make of all this? Despite the borrowings from Commedia dell'arte and the slightly dark underbelly of the play, The Taming of the Shrew still falls under the generic category of "Comedy." But, it's always a great idea for students to think about how plays can complicate and even deviate from categories of genre. (This is an especially good exercise for The Merchant of Venice and Measure for Measure, which are often identified as tragic-comedies or problem comedies.)