In As You Like It, Rosalind is banished by her uncle, who has also overthrown her father. Rosalind is smart and knows it is dangerous for a young woman to travel in the world. She decides to disguise herself as a man. This shows the power imbalance between men and women, but Rosalind shows us that women are not helpless. Rosalind discovers the power she has when presenting as a man. She is able to gain Orlando's confidence and gives him advice when hearing of his feelings for her (Rosalind, not Ganymede).
Like Rosalind, Hermia flees into a forest and attempts to take control of her own life. In A Midsummer Night's Dream, we see how men have power over women. Theseus captures Hippolyta, Queen of the Amazons. She is forced to marry him, although her portrayal at the end of the play suggests perhaps she is a willing bride instead of a victim. Hermia's father orders her to marry Demetrius, even though she and Lysander love each other. Egeus sides with Theseus, and so Hermia and Lysander run away together. When Lysander falls victim to Puck's spell, Hermia does not give up on him.
Katherine is not a typical woman and is labeled as a "shrew" because of her loud and boisterous behavior. She is not afraid to speak her mind. Petruchio makes a bet that he can "tame" her and does so by mistreating her until she listens to him. A battle of the sexes can be debated in this play, as we question Katherine and Petruchio's relationship.