The question of whether The Tempest is autobiographical has long been a source of speculation among scholars and literary critics alike. One of the main pieces of evidence offered in support of the autobiography thesis is the nature of the work itself. Previously, Shakespeare had almost always used preexisting source material as the basis for his plays. He always managed to transform such material into something wholly unique, but it was still preexisting material, all the same.
But this is not so with The Tempest. There is simply no precedent for the elements that Shakespeare, in a Prospero-like way, conjures up to create the magical world in which this, his final play, is set. The Tempest is all Shakespeare, conception as well as execution. And it's telling that when John Heminges and Henry Condell—fellow actors and friends of Shakespeare—compiled the first complete edition of his plays, they ignored the order in which the plays were written and put the last one, The Tempest, first. This could be taken to suggest that Heminges and Condell saw the play as not just the ultimate example of Shakespeare's original power as a dramatist but a unique expression of his personality.
There are strong links between The Tempest and William Shakespeare's own life. These revolve around the character of Prospero, his powers, and where he is in his "career" on the island.
Start with the basics: both are men of mature age. That's basic, but what follows is less basic.
Prospero is a wizard. On this island, people do what he says. They move according to his wishes. He organizes the drama early in the play, and the narrative twists and turns throughout it. At times, his magic makes them see things that aren't really there.
Shakespeare is a playwright, not a wizard, but on stage, people do what he says. Characters move according to his wishes. He structures the drama throughout his plays, and, when he moves into his poetic moments, like in the monologues, he makes us all see things that aren't there.
At the end of the play, Prospero sets his staff (his magic) aside. This furthers the comparison between Prospero and Shakespeare because The Tempest was Shakespeare's final play.