Symbolism, Imagery, Allegory
The play (including its title) is full of animal imagery, especially as it relates to the training and domesticating or hunting of birds and beasts. In the Induction, the Lord refers to Sly as a "swine" and a "beast" before transforming him into a "nobleman." Kate, of course, is referred to throughout the play as a "shrew," a derogatory term for opinionated and aggressive women that derives its name from a small, feisty animal. Punning on the name "Kate" and "cat," Petruchio threatens to turn Kate from a "wild Kate" to a "household Kate" (2.1.293). Once married, Petruchio aligns Kate with his "horse," his "ox," and his "ass," and later compares her to a falcon, a bird of prey he must starve and deprive of sleep in order to break or tame.
Literary critic Jeanne Addison Roberts points out that "Katherine is associated with more animal metaphors than any other female character in Shakespeare." (What's up with that?) Roberts also notes that "a great deal of the humor of the first meeting between Kate and her suitor […] depends on the determination of each to reduce the other to subhuman status."
OK, we'll buy that for a dollar but, so what? How can we turn Roberts's awesome observation into a snazzy interpretive point? (Some teachers call a snazzy interpretive point a thesis.) Well, let's see. For starters, even though Kate gets in some great pot shots during the initial meeting with Petruchio, he wins the game in the end and confirms man's dominion over beasts and, well, women. So, even while Petruchio sets out to elevate Katherine from a nasty little "shrew" to something more human – a nice, obedient wife – he simultaneously degrades her into an animal state.
There's a whole lot more to be said about the animal imagery and metaphors in The Taming of the Shrew. Go ahead and dig around in the play and see what you come up with. We're pretty sure we smell a great paper topic…