In act 1, scene 2 of William Shakespeare's Romeo and Juliet, Lord Capulet, Juliet's father, tells Paris, one of Juliet's suitors, that Juliet "hath not seen the change of fourteen years" (1.2.9). In fact, according to Juliet's mother and Nurse in act 1, scene 3, Juliet is a little more than two weeks shy of her fourteenth birthday. Lady Capulet states, "She's not yet fourteen" (1.3.14), and Juliet's nurse, who's been intimately involved with Juliet since the day she was born, states definitively that in a little more than two weeks, "Come Lammas Eve at night shall she be fourteen" (1.3.21).

There is no such direct evidence of Romeo's age anywhere in the play. It seems odd that Shakespeare makes an effort to clearly establish Juliet's age—an age which is notably younger than the average age for a woman to get married in Shakespeare's time—but Shakespeare never mentions Romeo's age.

In one of two major sources for Shakespeare's Romeo and Juliet, Arthur Brooke's narrative poem "The Tragicall Historye of Romeus and Juliet," published in 1562, Juliet "Scarce saw she yet full sixteen years: too young to be a bride!" (line 1860), but nothing is said about Romeo's age except "Upon whose tender chin, as yet, no manlike beard there grew" (line 54).

In the other major source for Shakespeare's play, the story of "Rhomeo and Julietta," novel 25, in William Painter's The Palace of Pleasure, published in 1567, Julietta is "a yong Gentlewoman of Verona," no age given, and Rhomeo, "of the family of Montesches," is "of the age of 20 or 21 Yeares."

Shakespeare's Romeo seems younger and more immature than Painter's Rhomeus of "20 or 21 Yeares." Nevertheless, Painter's Rhomeus, like Shakespeare's Romeo, falls in love with Julietta at first sight, impulsively marries her, kills Tybalt, is banished from Verona, goes to live in Mantua where he he's told, mistakenly, that Julietta has died, buys poison, and returns to Verona determined to kill himself at Julietta's tomb.

Back in Verona, Rhomeo finds Julietta seemingly dead in her tomb, whereupon he

drew the Poison out of his box, and swallowing down a great quantity of the same, cried out: "O Iulietta, of whole the World was unworthy, what Death is it possible my Heart could choose out more agreeable than that which yet suffereth hard by thee? what Grave more Glorious, than to be buried in thy Tombe? what more worthy or excellent Epitaph can be vowed for Memory, than the mutual and pitiful Sacrifice of our lives?"

Whether he's twenty-one years old, as in Painter's "Rhomeo and Julietta," or in his mid-teens, as he seems to be in Shakespeare's Romeo and Juliet, Romeo is still the same romantic, love-struck, immature, impetuous, passionate, and reckless "star-crossed lover" who, with Juliet, through their "misadventur'd piteous overthrows" and the "fearful passage of their death-mark'd love," by their deaths "bury their parents' strife" (Pro. 6–9).

Romeo and Juliet is the very famous story of the two "star-crossed lovers" (Prologue to Act I, line 6) who are doomed to fail in their attempts to overcome the feud which has dominated the Montague and Capulet families for as long as anyone can remember. They clearly underestimate the stubbornness of their parents and the hatred in their hearts which even extends to the servants who bear as much animosity towards the opposing household as they would towards a true enemy.

Romeo and Juliet's inability to grasp the true depth of feeling involved in this "grudge" (Prologue, line 3) gives an indication of just how young they both are. Neither Juliet nor Romeo appreciates the complexity of the problem and they even believe that denying their ancestry may be enough. Juliet ponders that "a rose by any other name would smell as sweet" (II.ii.44) suggesting that a family name should not have any bearing on her relationship with Romeo and she has come to the conclusion that it is "but thy name that is my enemy" (38). Romeo is also willing to renounce everything for her because as he says, his "name is hateful to myself" (55). Romeo is not yet old enough to understand that he would be expected to uphold the resentment and loathing in honor of his family name.  

The ages of Romeo and Juliet are significant in understanding how they come to such a tragic end because it is their youth and inexperience and their inability to see beyond their immediate circumstances that causes them, Romeo in particular, to act so irrationally. Romeo is so overwhelmed by his circumstances and so immature in his actions that he is apparently just a teenager of perhaps sixteen years old, definitely younger than eighteen because by eighteen a young man of his standing would be expected to be able to lead men into battle and so he would not act so impulsively. 

As robertwilliam states, Shakespeare never specifies Romeo's age. It is generally assumed he is a few years older than Juliet. The depiction of his essential youthfulness is what really matters in any case. He displays the ardent passion, the romantic idealism and also the rashness that one might say is natural to a youthful temperament.

It goes without saying that Romeo and Juliet have become the archetype of young, ill-fated lovers, attempting to breach the conventions of the adult world that surrounds them. The irony is, of course, that their elders and supposed betters display a lack of rationality and maturity in continuing their bitter feud, seemingly just for the sake of it. It is a pointless feud which ends in tragedy. Against such a backdrop, the youthful lovers appear to have better sense and judgement than their parents. This applies particularly to Juliet, who, as many have noted, displays a sense of dignity and shows a resourcefulness which would seem to belie her tender years. She is younger than Romeo but in some ways she appears the more mature of the two.

Good question. Juliet's age is, you're quite right, specified by the Nurse and Lady Capulet in Juliet's first scene as not-quite-fourteen.

As for Romeo's age, Shakespeare never tells us! We know - from what Friar Laurence and the Nurse comment on various occasions - that he is a "young" man (2.4.119) but we don't know precisely how young. A stage tradition has grown up of playing him as older than Juliet.

Shakespeare's sources don't really help either. Arthur Brooke's poem about Romeo and Juliet has Romeo's "tender chin" without a beard - suggesting, perhaps, anything from 12 - 18? And Bandello - whose novella Giulietta e Romeo was the source for Brooke's poem -  gives his age as 20 or 21.

So the answer is - we just don't know.