It is Katherine and Petruchio's wedding day, and Baptista, Gremio, Katherine, Bianca, Tranio (disguised as Lucentio), and Lucentio (disguised as Cambio) are all present for the ceremony. Petruchio, however, is late. Katherine complains that she is being married off against her will, that Petruchio is mad, and that he probably doesn't even intend to really marry her. She exits, crying.
Katherine and Petruchio's wedding is more than just a union of two people. It is an elaborate, social event and performance. Here we also gain some insight into Katherine's character: while she often eschews what other people think, she is still hurt by being humiliated in front of so many people.
Biondello enters and announces that Petruchio is coming, but he is dressed in bizarre and old clothes (including two mismatched boots) and is riding an old, feeble horse. His servant Grumio, accompanying him, is equally ridiculously dressed.
Petruchio and Grumio finally arrive. Petruchio acts as if nothing is strange about his attire, and asks where Katherine is. Baptista is offended by Petruchio's lateness and ridiculous clothing. He and Tranio tell Petruchio to change his clothes before seeing Katherine. Petruchio refuses and goes to find Katherine. Baptista pursues him.
Left alone, Tranio and Lucentio discuss their plan to get Bianca for Lucentio. Tranio tells him that he will find a man who will pretend to be Lucentio's father, Vincentio. This man will promise great riches to Lucentio, so that Baptista will approve of his daughter marrying him. Tranio assures Lucentio that he will beat out all of Bianca's other suitors.
The fake Vincentio offers another example of a deceptive false identity. Once again, Baptista's main concern in finding a husband for Bianca is money.
Gremio enters, having just come from the church where Katherine and Petruchio were wed. He calls Petruchio "a devil, a devil, a very fiend," (iii.2.157). Tranio counters that Katherine is a devil, herself, but according to Gremio, she's "a lamb, a dove, a fool to him," (iii.2.159). Petruchio acted rudely at the wedding and swore loudly in front of the priest who was marrying the couple. He drank raucously, made a mess, and kissed Katherine "with such a clamorous smack / That at the parting all the church did echo," (iii.2.180-181).
Gremio describes Petruchio's elaborate performance at the wedding (though it is unclear how genuine some of his bad behavior might be). The same strong-willed qualities that make Katherine a shrew make Petruchio "a devil", and perhaps make him a good match for Katherine. Though note how Petruchio is partially admired for his behavior, while Kate decidedly is not. There is a double standard at play here.
Petruchio enters, along with Katherine. Bianca, Baptista, Hortensio, and Grumio. Petruchio announces that he will leave now, skipping the traditional wedding feast that has been arranged. Tranio and Gremio ask him to stay, but Petruchio insists that he must leave, offering no explanation for why he cannot stay. Katherine asks him, "Now, if you love me, stay," (iii.2.209) but Petruchio refuses.
Petruchio begins to exercise his authority and control over Katherine, now that they are married. Katherine asks him to stay at the feast out of love for her, but Petruchio shows that, for him, marriage is more about power than love.
Katherine says she will not go with Petruchio, but rather will enjoy the feast without him. Petruchio insists that she come with him, calling her his property and saying, "I will be master of what is mine own," (iii.2.235). He acts as if he is protecting Katherine from her family and takes her away, leaving with Grumio.
Everyone is astonished at Petruchio's behavior. Bianca says that Katherine is mad and has found a fittingly mad mate. Baptista tells everyone that they can still enjoy the feast, and that Lucentio (actually Tranio) and Bianca can take the places of Petruchio and Katherine.
It is still unclear how much of Petruchio's bizarre behavior is an intentional performance, an act put on to tame Katherine, or whether he is actually this difficult. From Bianca's point of view, Katherine and Petruchio make a good match, since they are both so stubborn and mad.