Shakespeare is the most famous playwright of all time, and The Taming of the Shrew is one of his most well-known comedies. It's also the most controversial, mainly because of the much-debated misogyny/not misogyny at its center.
For those unfamiliar with the plot, here's a refresher: Katherina and Bianca are both daughters of a nobleman. While Bianca is well-loved and sought after by many suitors, Kate is a "shrew," and her aggressive, stubborn disposition drives away any potential suitor's affections. A man named Petruchio is offered financial rewards if he will marry her and tame her. So here goes. The two constantly spar, but Petruchio breaks Kate down through different psychological torments until she is tamed and expresses her deference to him.
To a contemporary reader or audience, this immediately raises red flags. Not only does it cause friction with a more contemporary view of women's rights and gender roles, but it's also incongruous with many of Shakespeare's works.
Although I don't like to apply modern terms like "feminist" to someone who lived centuries ago, a running theme of Shakespeare's comedies is that the female protagonists end up with the person they want to be with. The women are usually empowered in their love lives, so it seems shocking to see it flipped on its head here.
In most productions of The Taming of the Shrew, the creators seem to try to flip the play back to a place where the audience will be more comfortable. They don't want to play Petruchio as an outright misogynist, or Kate as someone who is truly beaten into submission, so creative teams often attempt to make the play seem more like Shakespeare's other comedies.
You can say that Petruchio truly loves Kate and wishes to get closer to her--well, it might explain away a few lines, but it will hardly change the fact that he keeps her from eating and drinking. You can say that Kate really loves Petruchio and is just afraid of letting her walls down, but even if she does love him, it doesn't change Petruchio's actions. George Bernard Shaw said about an actor playing Petruchio as a sympathetic character: "He cannot make the spectacle of a man cracking a heavy whip at a starving woman otherwise than disgusting and unmanly."
Perhaps in an effort to visibly lessen some of the misogynistic themes and actions in The Taming of the Shrew, some productions have switched the gender of the characters, or cast the show with actors of all the same gender. But you don't need to look outside the play's text to see a gender-bending version of the story--although few productions show it, it's right there, before Kate and Petruchio's story begins.
The Taming of the Shrew truly opens with the story of Sly, a drunken peasant. A lord happens upon Sly, sleeping in a drunken stupor, and decides that it would be fun to see if he could convince Sly that he was actually a great lord, with servants and a noble wife. He dresses Sly in fine clothes, has his servants attend him, and has his page dress as a woman, pretending to be Sly's wife. Kate and Petruchio's story is presented as a play for Sly's entertainment.
And all the while the servants, page, and lord insist, despite Sly's arguing, that Sly is a great lord until Sly's trust in his own mind breaks down and he accepts that he is something that he is truly not. Sound familiar?
When you see these two stories next to each other, it feels harder to argue that Kate and Petruchio's relationship is secretly loving and healthy, that perhaps Kate pretended to be tamed so Petruchio could win a bet, or that maybe he treated her respectfully in secret. Both Sly and Kate's stories show a character being worn down until they accept whatever is told them, although Sly's taming is less disturbing than Kate's.
This doesn't mean that The Taming of the Shrew isn't relevant for a modern audience. It just means that it can't be treated like Twelfth Night or A Midsummer Night's Dream. Despite the entertaining drama, and yes, the comedy that exists in the exchanges between Petruchio and Kate, there are elements of tragedy to this story.
Whether or not you believe the characters truly end up happy at the play's conclusion, the fact remains that more than one character has been changed by a person exerting physical and psychological power over them. Shakespeare shows us there's humor in "taming" another human being, but we're pulling wool over our own eyes if we think he isn't also showing us the damage.