A romantic comedy can be defined as being one
that presents the adventures of young lovers trying to overcome social, psychological, or interpersonal constraints to achieve a successful union.
As You Like It certainly fits this definition. Orlando and Rosalind are unknown to each other at the beginning of the play. Orlando's victory in the wrestling match, which starts as a reason for celebration in his eyes, becomes an obstacle when he and Rosalind become aware of the complications it presents to their futures.
What is thy name, young man?
Orlando, my liege; the youngest son of Sir Rowland de Bois.
I would thou hadst been son to some man else.
The world esteem'd thy father honourable,
But I did find him still mine enemy:
Thou shouldst have better pleas'd me with this deed
Hadst thou descended from another house.
He calls us back: my pride fell with my fortunes:
I'll ask him what he would.--Did you call, sir?--
Sir, you have wrestled well, and overthrown
More than your enemies.
And so the stage is set. Orlando and Rosalind, with Celia and Oliver supporting the intrigues of their friends. The complications and mistaken identities add to the comedic aspects of the play, while the romantic aspirations of all the characters drive the continuation of the action.
Shakespeare recognized the draw that romantic comedy had for the audiences that would be attending his plays, and As You Like It continues to be one of the most frequently producced of his works.