Juliet's destiny is to die a tragic death by her own hand. We know this because it explicitly says so in the play's Prologue. It's all in the stars, apparently, an indication of the widespread belief in astrology that characterized the age in which the play was written.

What we don't yet know, of course, is how this all comes about and why. There's an awfully long way to go before we reach that heart-rending moment when a distraught Juliet will stab herself to death upon finding the corpse of her beloved Romeo. That gives Shakespeare plenty of time to devise a suitably intriguing and convoluted plot that brings the star-crossed lovers together forever, but only in death.

It's a measure of The Bard's considerable skills as a dramatist that we're emotionally invested in Romeo and Juliet, even though we know of their tragic fate. There always seems to be the faintest hope that maybe, just maybe, the two young love-birds will avoid their unhappy fate. That such hope exists at all is a testament to the immense skill with which Shakespeare draws his two main characters and of the rich, intriguing plot in which he embroils them.

We learn in the Prologue to Romeo and Juliet, delivered by the Chorus, that the love affair that blossoms between the title characters will not end happily for them:

A pair of star-cross'd lovers take their life;
Whose misadventure's piteous overthrows
Doth, with their death, bury their parents’ strife.

"Star-cross'd" here means something more than "unlucky." It basically means that Romeo and Juliet are doomed to die by fate. So their destiny is to meet, to fall in love, and to die as a result of their love. At the same time, their deaths will "bury their parents' strife." In other words, the feuding Capulets and Montagues will be reconciled by their grief. It took the deaths of these two young lovers to make them realize how destructive their mutual hatred really was. 

Some might think that discovering that the two lovers are destined to die in the Prologue lessens the suspense of the play. But it also creates a sense of dramatic irony that is absolutely heartbreaking. Romeo and Juliet's love is tender, genuine, and eloquently expressed, and knowing that it is doomed makes their declarations of love and of hope all the more poignant. We know that they cannot escape their destiny, yet we find ourselves rooting for them until they meet their tragic end together.