The Taming of the Shrew is a comedy written by William Shakespeare, and set in an unusual play-within-a-play format. The play opens in a tavern where Christopher Sly is behaving in a drunk and disorderly fashion. Upon the arrival of a Lord and a troupe of travelling players (actors), Sly is convinced (as a joke) that he is a great Lord and the players put on the play that ensues for his enjoyment. This is called the Induction of the play and is often not performed.
The plot of the play that Sly is audience to concerns a man of Padua (the "second" setting of the play) named Baptista Minola and the trials and tribulations of trying to get his elder daughter, Katharina or Kate, married before he can marry off his much more desirable younger daughter, Bianca. Kate is considered not be marriage material by the local bachelors because she is outspoken and shrew-ish in behaviour.
A suitor, Petruchio, does arrive and make a contract with Baptista to marry Kate. He whisks his new bride of to his home in Verona (the third setting), where he "tames" Kate into a good and obedient wife. Kate's younger sister Bianca is also wed to her love, Lucentio by the end of the play.
Much of this play's comedy comes from physical humor-- slapstick interactions between Petruchio and his servant, Petruchio and Kate, and between other servants. The play also features characters in disguise which leads to another favorite comic device of Shakespeare's -- mistaken identity.
This play is often considered a "problem" play by scholars, since it isn't 100% clear whether Petruchio has in fact, by the play's end, tamed Kate. Theatrical productions of this play often make bold, definitive choices "for" or "against" this conclusion, and these staging decisions can have a large effect on the resolution of the play's main conflict: Can Kate be transformed from a "shrewish" woman and wife, to an obedient and loving one?
For more on The Taming of the Shrew, please follow the links below to the Enotes Study Guide.