Shakespeare's play may have been inspired by a real life shipwreck. In 1609, a ship named the Sea Venture (part of the Virginia Company) was on its way from England to Jamestown (yep – that's the early American colony alright) when it wrecked in the Bermudas. The crew was thought to be lost forever but managed to survive on a deserted island. So, does this mean The Tempest is inspired by the New World? Check out this "Shakespeare in American Life" podcast if you want to know more...
The title of Aldous Huxley's Brave New World comes directly from Miranda's speech "O brave new world, that has such people in it!" There's even a character in Brave New World who can't stop talking about Shakespeare.
BBC News Broadcasting House has a statue of Ariel and Prospero on the front of the building. It's rumored that "on initial inspection by the BBC's governors, they demanded that several inches be lopped off the manhood of the sprite Ariel" (source).
Patrick Stewart, known as Captain Picard of Star Trek and Professor Xavier in the movie versions of X-Men, has played Prospero on stage (source). We can totally see that.
Musician Beck's 2002 release, Sea Change, was ranked 440 in Rolling Stone's list "Top 500 Albums Of All Time." The term "sea change" originated in The Tempest, with Ariel's "Full fathom five thy father lies" speech, and has come to mean an incredibly significant change (source).