The primary external conflicts in The Taming of the Shrew occur between Katherine and Petruchio, Katherine and Bianca, Katherine and her father, and, to an extent, Lucentio and Baptista.
Katherine and Petruchio quarrel repeatedly because she is the titular “shrew” and he is a brash, unfazed man who is not scared to challenge Katherine. This verbal sparring continues until Petruchio takes Katherine to his house, where he employs various methods to “tame” her. Katherine has a conflict with her father, Baptista, because he wants her to marry, but Katherine refuses to change her behavior to attract a suitor. Katherine’s conflict with her sister is based on the fact that Bianca, despite attracting multiple suitors, is forbidden from courting until Katherine is married. Finally, Lucentio’s indirect conflict with Baptista stems from his forbidding Bianca to pursue marriage. To solve this external conflict, Lucentio devises a plot to find a man who would marry Katherine.
The primary internal conflict in the story, though not directly explored, is Katherine versus herself. She desperately wants to marry and be happy, as evidenced by her dismay when Petruchio is late to their wedding. Despite this, Katherine also has a difficult time controlling her rage, and it frequently gets in the way of her happiness. When Petruchio arrives and is able to match her in vileness and insult, Katherine perhaps marries him because she believes he is capable of helping her overcome her internal conflict.