What is Romeo's age?

There is no way to know for certain what Romeo's exact age is, as it is never explicitly stated. We know that Juliet is 13, soon to be 14. Though the age of consent in Elizabethan times was 12 years old for a woman, some critics have accused Romeo of having a tendency towards pedophilia, given that his preoccupations, friends, and activities make him seem anywhere from 18 years old to his early twenties. However, his relationship with his friends—particularly given their tendency to poke fun at him over his constant love-sickness—may indicate that he is something of a "little brother" to characters like Mercutio. It is also stated that Paris is older than Romeo, and he is at a point in his life when it is expected for him to be married. It is for these reasons that one might place Romeo, as many critics have, around 16 years old.

Romeo's age is not given in the play, but there are many hints he is a teenager. First, Juliet, his beloved, is not quite 14. Second, Paris, the man Juliet's father betroths her to, is older and established, so at least 25. Romeo seems to be younger than his rival, so probably no more than 20.
Romeo, like Juliet, still lives in his father's house and doesn't seem to have any sort of career or responsibilities. His father worries as the play opens because Romeo is up all night mooning about over Rosaline. This makes Romeo sound very much like a teenager. 
Romeo's impulsiveness also seems very characteristic of a teenager. He hasn't yet learned to take the long view. He simply doesn't, for example, want to wait for marriage to Juliet. Instead, he wants to marry her the day after meeting her. A more mature individual would probably want to get to know his prospective mate a little better before rushing headlong into marriage.