There is a theory that Kate doesn't genuinely submit to Petruchio but is putting on an act and merely becomes shrewd to get her way with her husband. Supporting this is how Kate doesn't gradually become submissive but, almost in exasperation, just starts agreeing with him in a completely unrealistic way, and this behavior gets Petruchio to do whatshe wants. (Thus learning the very lesson he's trying to teach: one catches more flies with honey than with vinegar.)
Another one: Petruchio is lampooning society (specifically gender roles) throughout the play; the "taming" is really him trying to get Kate to play along with him without having to drop the joke by telling her in front of other people.
Another: Petruchio's act is meant to show Kate how ridiculous her behavior must seem. She eventually catches on to this and the rest of the play is a prank on everybody else.
Yet another, for the last monologue, though it requires some side gags: Kate gets in on the bet and delivers the last monologue to get her and Petruchio the money.
The alternate alternate character interpretation is that Shakespeare means what he says and that attempts to read the play as "subversive" are a product of the modern audience's discomfort with the story.
And for that matter, the alternate alternate alternate character interpretation (whew!) is that Shakespeare was being sexist but Fair for Its Day, because Kate's last speech does not say only "wives submit to your husbands" but rather "wives submit to your husbands because they have your best interests at heart"
Another Fair for Its Day interpretation would note that there were other contemporary plays in this genre (taming a bad wife), only they solved the problem by beating the wife into submission. You know, comedy! Shakespeare's version doesn't seem that bad in comparison...
Yet another one is that Kate intended to be a "good" wife all along, but just wanted a strong husband. Once she realized that her act wasn't scaring Petruchio away, he earned her respect.
Gregory Doran's 2003 RSC production played Petruchio as a genuine madman driven to distraction by his father's death and Kate as a troubled woman who loves him enough to accept him the way he is, turning the whole play into a story about mutual support under difficult circumstances.
In terms of more minor characters, the 1992 RSC production drastically reimagined Tranio as a manipulator trying to steal Bianca for himself. In the end Lucentio wins thanks to some words of warning from Biondello, but a Bittersweet Ending ensues as Bianca still loves Tranio.
Surprisingly, both male and female readers, viewers and audience members have viewed Kate as a woman who needs to learn humility. To be fair, they have a point; at one stage Kate hits a music teacher over the head with a lute simply because he told her she played some notes wrong, and at another stage she strikes Petruchio just for making a crude - but fairly benign - joke about her
Yet another alternative: Petruchio is playing a game of contract chicken with Kate. Back in the day, the standard marital exchange was that the man would provide for and protect his wife, and the woman would respect/obey her husband and put out for him. Kate is refusing to show respect for Petruchio, so he is simply declining to fulfill his side of the "contract". In his estimation, it's only a matter of time before she blinks, and she will before he does. Once she shows signs that she understands it now, he softens considerably toward her.
There is also another theory (link here) that Kate and Petruchio are doing a rudimentary BDSM routine. It helps that immediately after Kate finished her final speech about being a slave and submitting utterly (Does This Remind You of Anything?), they went straight to bed.
Tranio doesn't hesitate a bit to go along with Lucentio's plan to wed Bianca. Undying Loyalty between a servant and master who have been friends since boyhood? Or is he merely a crafty Servile Snarker who sees in his boss's marriage to a wealthy girl a chance for a nice payday? Some productions have even played Tranio as in love with Lucentio, making his willingness to please his master even more interesting...
Kate sometimes comes across as a bully to her younger sister. This results, in what appears to be, Petruchio giving her a taste of her own medicine.
The 2015 Stratford Shakespeare Festival's performance featured a rather... brutal Kate. From chasing a Bianca with bound hands with a sword to bashing Bianca's face into a pillar, she really does deserve what she got.
That brutality wasn't spared on Petruchio either. There was some pretty physical wrestling going on between the two during the meeting scene. We're talking full-on headlocks and leg holds, like she was trying to crush his head between her thighs while hanging sideways off his neck.
Another rather more sinister possibility is that Kate actually believes what she's saying becase she's suffering from some sort of Stockholm Syndrome and that Petruchio's "taming" methods look very much like now-recognized forms of psychological torture.
It is also possible to read Kate's final speech as (a) a Take That to women who have been making her life miserable for years by vaunting her superior wifely virtue. And (b) really intended for Petruchio. The burden of her speech is that men love their wives and work for their benefit, she is telling her new husband that she understands why he's behaved the way he has to her and accepts, even welcomes, the lesson and his affection. The message is received: "Kiss me, Kate!"
There is also an interpretation which plays up the Framing Device where the story is actually a play in-universe, and one presumably being performed by male actors for a male audience. In this interpretation, the fact that no real woman has ever responded to things in the way that Kate does is the point, and the joke is that the play is absurd but the guy it's being put on for is too dumb to notice.
Angst? What Angst?: Played for Laughs in the Induction. When the Lord's servants tell Christopher Sly that he's been "in a dream" for fifteen years, he responds, "Fifteen years? By my fay, a goodly nap!" Upon being told that everyone he knew never existed, he adds, "Now, Lord be thanked for my good amends!"
Fair for Its Day: Many complain about the ending, but in historical context Shakespeare was writing in a time when women were just beginning to be treated like human beings rather than the property of men. The character of Kate was pushing the boundaries of what was acceptable in society, so some agree that Shakespeare added the themes of female submissiveness to appease his audience.
Rule-Abiding Rebel: The play is praised by some as a proto-feminist work. Kate's speech at the end is taken by them to be ironic. It is likely not.
Unintentionally Sympathetic: even some of Shakespeare's contemporaries felt the treatment of Katharina was a bit harsh. This is even more prominent among modern readings, with Kate being popularly interpreted as an abuse victim.
Values Dissonance: Even the most subversive and proto-feminist interpretations can't make all of the sexism palatable to modern audiences. And if you don't read it as subversive—if you take the text at face value, as a story of a domineering man breaking a woman to his will and turning her into a submissive — the Values Dissonance is cranked Up to Eleven. (As noted above, many of the "taming" methods look a lot like what we'd now call psychological torture to some interpreters.)
The dissonance may not be restricted to just the modern day, though. It only took ten years for there to be written an unofficial sequel to the play called The Woman's Prize (The Tamer Tamed), where Petruchio gets a helping of his own medicine by his second wife Maria.
Values Resonance: Modern series such as the Fifty Shades and Submissive trilogies (written in the 21st century) also tell the story of a domineering man turning a woman into his submissive, and some modern female readers apparently love them.