The cinematic history of this play has been as wild as that of any of Shakespeare's works -- the net result is a remarkable diversity of production. The four cited here are merely those most widely available, but even they represent quite a remarkable spread.

1929, Sam Taylor: In 1929, Sam Taylor directed the legendary Hollywood couple Douglas Fairbanks and his wife Mary Pickford in their only production together. Their The Taming of the Shrew was reasonably well-received at the time; since then its reputation has been in almost continuous decline, and is now regarded as a fairly bad effort. I have not yet had a chance to see it, and so will reserve judgment myself, but it seems likely that it would be interesting just as one data-point among several, showing how Shakespeare was received and handled in different periods, forming part of a broader history of taste.

1967, Franco Zeffirelli: At the height (or depth) of Elizabeth Taylor's brief and stormy marriage to Richard Burton came this tumultuous but exceedingly robust -- even athletic -- performance of The Taming of the Shrew. Hollywood pundits were quick to point out how the play did (or did not) reflect the fortunes of their troubled household.

Gossip-mongering aside, however, the play is a mix of good and bad. Taylor and Burton are not really well-matched: Burton was a stage-trained Shakespearean of enormous range, while Taylor was an actress of fairly limited capacities. Burton's diction is superb; Taylor's is adequate at best. But the volatility of the personalities and Zeffirelli's imaginative direction impart to this a rather giddy energy that makes up for a great deal, and provides a very congenial introduction to Shakespeare for those who have not seen his plays on stage or film before.

The humor is largely physical, and of a slapstick sort; this does not depend especially on the disparity of diction in the principal actors. As always, Zeffirelli manages to bring interesting secondary characters into his production: Michael Hordern is the quintessential old man, out of his depth and a bit intimidated by his two daughters. Michael York makes an appearance -- not the last he would do for Zeffirelli (he went on to do Tybalt in Romeo and Juliet and a remarkably convincing John the Baptist in Jesus of Nazareth.)

1976, Kirk Browning: This film (now available in the Broadway Theatre Archive series) is a recording of a stage production by the American Conservatory Theatre of San Francisco, with Marc Singer as Petruchio and Fredi Olster as Kate, filmed before a live audience. It is cast unabashedly in the mold of the Italian Renaissance Commedia dell' Arte: it's slapstick throughout, with non-stop sight gags, general silliness, and a certain amount of bawdy humor.

Declamation is exaggerated and rapid-fire; all the actors prance around the stage with peculiar leaps and comic pauses; every major action is punctuated with percussion rachets, bicycle horns, and rim-shots, and the interaction of Petruchio and Kate is extraordinarily athletic, with an almost circus-like complexity of physical humor. Be forewarned: some of it is rather crude, but the timing is perfect, and the interaction of the leads is electric.

1980, Jonathan Miller: Almost at the very opposite end of the spectrum from Zeffirelli's and Browning's renditions is this entry into the BBC Shakespeare Plays series. Reactions to it were very mixed, and a number of people vehemently disliked it.

Costumes and staging are minimal, but that's only the beginning. The heart of its strategy is that it casts John Cleese of Monty Python fame (quite a capable actor, in fact) as Petruchio. Rather than the swaggering conqueror one finds in most productions of Shrew, he plays a role rather like his nameless character in The Great Muppet Caper, who watches Miss Piggy scaling the outside of his house with a kind of bemused detachment. Here also he hems his role around with a very drily humorous English reticence, appearing to be perpetually perplexed, but at the same time wholly unrattled by anything that happens. In the end it is his apparently indestructable imperturbability that wears Katherine down. Doing this to the character of Petruchio is in most ways counter-intuitive, but the effect is (to my taste) oddly appealing. Whether one likes the film or not, however, it forms, in conjunction with the Zeffirelli production, almost a case study in how flexibly one can approach Shakespeare without transgressing the basic sense of the text.

1988, Richard Monette: This is a production from the Stratford (Canada) Shakespeare Festival, and played on a very Shakespearean thrust stage, like their productions of Romeo and Juliet and As You Like It. Overall, it is quite solid, though the tape quality of the only available release is third-rate. The experience is more theatrical than most others, withe the possible exception of the remarkable but rather extreme 1976 version.

Related:

Cole Porter's Kiss Me, Kate is a split-level treatment of The Taming of the Shrew, in that it purports to be about a production of the play, while at the same time having its own problems of marital harmony. The play is also the source for the cheerfully thematic song, "Brush Up Your Shakespeare". Two cinematic versions of it are available:

1953, George Sidney The film version with Howard Keel and Kathryn Grayson is very watchable, though it is a far cry from the stage play, which is a good deal tighter and more disciplined. It does feature some very impressive dancing by Ann Miller.

2003, Michael Blakemore: This is a filmed version of the recent Broadway revival of the play.

One famous episode of Moonlighting, ("Atomic Shakespeare") is based very broadly on The Taming of the Shrew. It abuses the language rather frightfully, and completely subverts the basic ideas of the play, but it is rather funny in places.

More recently, the high school comedy "10 Things I Hate About You" offered a permutation on The Taming of the Shrew. It was neither intensely bad nor particularly good.