Captain Jack Harkness, the Doctor and Martha Jones
My biggest complaint with this episode is that it's largely filler - it's a way of re-introducing Captain Jack, whom we haven't seen in "Doctor Who" since the end of season 1, and of introducing The Master to New Doctor Who. So for the first thirty minutes or so, not a lot happens. The episode opens with the Doctor and Martha landing the TARDIS on the time rift in Cardiff to refuel (something the Ninth Doctor did in season 1's "Boom Town). Captain Jack appears in the distance, running madly towards the TARDIS, shouting for the Doctor. The Doctor sees him on the view screen in the TARDIS, but ignores him and the TARDIS dematerialises - but with Jack clinging to the outside, which propels the TARDIS forward in time thousands of years, and in space, to the very edge of the universe, where the last remnants of humanity are still clinging to existence. Outside their compound exists a race of mutated humans who are vicious, savage and enjoy hunting down (and presumably eating) regular humans. The TARDIS lands and Martha asks the Doctor what's out there. He admits he doesn't know, and she asks him to repeat that since it's rare that he doesn't know where they are (or anything else!) After the Doctor suggests to Martha that they should really leave, the pair hurry outside where Martha spots Jack lying on the ground nearby, apparently dead. She dashes back inside the TARDIS whilst the Doctor says greets Jack less than enthusiastically. Just as Martha is telling the Doctor that Jack is dead, he springs back into life, scaring her silly, although she soon recovers when he flirtatiously introduces himself. Martha's a bit shocked when the Doctor reveals that he knows Jack and that he used to travel with the Doctor.
We then cut to Jack explaining to Martha what happened to him (something "Torchwood" viewers waited 13 weeks for in vain) - that he woke up alive on Satellite Five after being exterminated by the Daleks and he used his Vortex Manipulator (a watch-like device he wears on his wrist) to get back to Earth. He sarcastically notes that the Doctor's not "the only one who can time travel", prompting the Doctor to reply "Oh excuse me! That is not time travel. It's like I've got a sports car and you've got a space-hopper" and Martha to comment "Oh-ho! Boys and their toys!" Martha then asks the Doctor if he makes a habit of abandoning his Companions in odd places around the universe and Jack makes a snide comment about "Unless you're blonde", prompting Martha to say (of Rose) "Oh so she was blonde". This causes the Doctor to lose his temper with them both, pointing out that they're "at the end of the Universe. Eh? Right at the edge of knowledge itself and you're busy - BLOGGING!"
Moments later they spot a human being chased by a group of humanoids (the vicious futurekind as they've been dubbed) and the three go haring to the rescue, only to find themselves even more out-numbered than they'd realised, and their route back to the TARDIS cut off as well. The man whom they've attempted to rescue suggests they ruin for the silo and the four hare off with the futurekind in hot pursuit. Once inside, someone tells Professor Yana (Sir Derek Jacobi) via an intercom that four humans have arrived and one of them's a doctor. He gets excited and rushes down to meet the four. He rushes the Doctor back to his lab and starts talking about the technology he's been trying to use to send a rocket out to Utopia which humanity is desperate to reach. Unsurprisingly, the Doctor steps in, and despite knowing nothing of the technology, helps out and gets it working in no time.
Up until this point, not much has happened, except that Professor Yana reveals that all his life he's had a noise in his head - the sound of drums - which has been getting louder as if they're getting closer. Then one of the futurekind, who has at some point snuck inside the compound, causes some havoc, which means the power fails and the radiation in the room where the rocket couplings are being prepared for the take off, reaches critical - and there's no way of restoring the power or lowering the radiation levels quickly. So the Doctor volunteers indestructible Captain Jack (who, since Rose's actions in looking into the heart of the TARDIS, taking the power of the Vortex into herself and restoring Jack to life, cannot die), to enter the room below the rocket so that he can deal with the couplings. He and the Doctor then discuss Jack's situation (one on either side of a door) and their conversation is overheard by Martha, Professor Yana and his assistant Chanto (a blue alien who looks like a humanoid bug). Bits of the conversation start to echo through Yana's brain and then Martha discovers that he has a fob-watch which is very similar to the one the Doctor had in "Human Nature" and "Family of Blood", which the Doctor used to contain his Time Lord essence whilst he was a human. Yana explains that the watch doesn't work and can't be opened, explaining that he's had it all his life, ever since he was found as a naked child on the shores of the Silver Devastation. Martha realises the watch's significance and runs to tell the Doctor about it, but whilst she's gone Yana opens the watch, restoring his true self. And his true self is revealed to be The Master, the Doctor's greatest enemy.
But the episode doesn't end there. The Doctor, Jack and Martha race back to Yana's lab, but they're too late. The Master has attacked Chanto, intending to kill her, revealing that he's never really liked her (although that's really the Master, not the Professor speaking) and expressing anger that in the 17 years she's worked with him, she never thought to ask him about the watch, which he's barely paid attention to as it's had a perception filter on it - just as the Doctor's did. He picks up the Doctor's hand (which had been chopped off by the leader of the Sycorax way back in "The Christmas Invasion" (Tennant's first episode as the Doctor) which Jack had kept stored in a jar at Torchwood Three to act as a "Doctor detector") and takes it aboard the TARDIS. Just as the Master is preparing to leave, Chanto shoots him and the Doctor and co. arrive - having had to fight their way through two locked doors and race against the futurekind (whom the Master had allowed into the compound).
After Chanto shoots him, the Master is forced to regenerate (from Derek Jacobi into John Simm) and he has a quick conversation with the Doctor, tauting him from the safety of the TARDIS (which he's locked from the inside) before he disappears with the TARDIS, leaving the others trapped at the end of the Universe.
And leaving the viewer wondering how they will get back to Earth - and just what plans Harold Saxon (the Master) has for Earth when he's elected as Prime Minister...
Professor Yana and the Doctor