Short answers, pulled from the story.
The play emerged from a foggy period between 1590 and 1592 when William Shakespeare first arrived in London. A Stationers Register entry for A Shrew on the 2nd of May 1594 provides a hard deadline for when the original must have been written.
Scholars struggle to pin down the exact date because the play exists alongside another anonymous work called A Shrew which shares almost the same plot but uses different names for characters like Bianca and Katherina. Some experts believe A Shrew is a bad quarto meaning actors tried to reconstruct Shakespeare's original script from memory after seeing it performed while others argue that both plays were based on a lost third source known as Ur-Shrew.
Katherina Minola stands defiant against any man who tries to court her while her father Baptista decrees that Bianca cannot wed until Katherina finds a husband first. Petruchio enters from Verona accompanied by his manservant Grumio to pursue Katherina so that Bianca becomes available for other suitors like Lucentio and Tranio.
George Bernard Shaw wrote a letter to the Pall Mall Gazette dated the 8th of June 1888 calling the play one vile insult to womanhood from start to finish. Critics have reacted with heartily supportive or altogether disgusted responses since its first appearance between 1588 and 1594 with some arguing the play contains a crudely reactionary dogma of masculine supremacy.
Cole Porter created the musical Kiss Me Kate which became one of the most famous adaptations of the play for stage and screen. McLintock! appeared in 1963 as an American Western comedy film starring John Wayne and Maureen O'Hara while Elizabeth Taylor and Richard Burton starred in the 1967 film version directed by Franco Zeffirelli.