season 1 of the rebooted Doctor Who back in 2005, the Face of Boe is a highly advanced, seemingly ancient being who doesn't have a body, instead resembling a giant floating head in a tank. In his first appearance, "The End of the World", he was sponsoring a viewing of the Earth's destruction, billions of years into the future, before making further appearances in the episodes "New Earth" and "Gridlock".
Another recurring element of Doctor Who introduced in season 1 was Captain Jack Harkness, an immortal time-traveller who enjoys to flirt with the Doctor (and just about anyone else too) and is much more of a dashing action hero type figure. Harkness may be immortal, but he is aging, albeit incredibly slowly, and Doctor Who has repeatedly drawn a link between him and the Face of Boe. The key line in this comes in Doctor Who season 3 finale "Last of the Time Lords", where Jack not only expresses concern over what he'll look like if he lives for millions of years, but also reveals that he was once nicknamed 'The Face of Boe', since he came from the Boeshane Peninsula. In "Gridlock", meaning, the Face of Boe refers to the Doctor as an "old friend".
While it may seem quite obvious to fans that Jack Harkness is turning into the Face of Boe, Doctor Who has avoided outright confirming it. That's a decision that stems from former showrunner Russell T. Davies, who introduced both Harkness and Boe into Doctor Who. In the commentary for "Last of the Time Lords", Davies refers to the idea as a "theory". Later in an interview with NJ.com, Davies explained his reasoning behind not making the connection between the two explicit: "...It's couched in that are not absolute gospel. There are these spin-off books and comic books, and every now and then I'll see a script for one where they say definitively that he's the Face of Boe, and I always stop those from being printed. I have my own personal theories, but the moment it became very true or very false, the joke dies."
Among Harkness' many defining traits is his vanity, with the character obsessed with his own good looks. The idea of Captain Jack turning into the Face of Boe, then, is a fate worse than death for him, which is what makes it such a great idea he will end up that way (although tinged with sadness for him as well). Still, Davies' reason for not wanting to confirm Harkness is the Face of Boe is understandable, although it isn't entirely out of the question that it could be made official sometime in Doctor Who's future.
The Face of Boe lives for billions for years, and Jack Harkness is immortal, so there's a long stretch of time for it to happen, and it's a story Doctor Who could eventually decide to show. Captain Jack's season 12 return hinted at his Face of Boe transformation, with his reference to nanogenes suggesting something within his make-up was gradually changing. It's a compelling story to tell and no doubt one fans would like to see on screen, and with Harkness finally back in the fold there's at least more of a chance of it happening. If nothing else, Doctor Who can still have a lot more fun building the links between Captain Jack Harkness and the Face of Boe, even without confirming it.