For years, Zoe Heriot has been having terrible dreams about monsters. She's supposed to have an eidetic memory, so the fact that she can't remember her dreams clearly implies that something has imposed blocks on her memory. Thus, she's taken her dreams to a counsellor, someone who might be able to explain why she's been dreaming about travelling through time and space with the Doctor. Zoe clearly remembers parting company with the Doctor and Jamie after the Cyberman invasion of the Wheel, but in her dreams... she stows away aboard the Doctor's TARDIS in search of adventure. The Doctor tries to dissuade her by using a thought scanner to show her his final encounter with the Daleks, but the inhuman killers frighten her more than he'd anticipated. He thus backtracks apologetically, assures her that the Daleks have been utterly destroyed, and concedes that the TARDIS could always use another genius aboard. Jamie shrugs and accepts Zoe as their new travelling companion, and the Doctor sets the TARDIS in flight.

The TARDIS materialises in Lavonia, a city built inside a protective dome on an asteroid. Zoe is fascinated by the myriad of alien life forms and cultures in the marketplace, and as she and her new friends explore, they see a giant news screen reporting that the neutral asteroid city is playing host to peace talks between the Xantha and the Tibari. The Xantha Empire is an independent former Earth colony, and the Tibari Republic was founded by an amphibious race of fish-like humanoids. Fifty years ago, the Xantha invaded the Tibari homeworld, and after decades of war, the two species have finally been brought to the bargaining table.

The Doctor and his companions are unexpectedly arrested on the spurious charge of fouling the pavement, but the offended Doctor realises that the guards are not Lavonian police but Xanthan soldiers. Since they're armed, however, he and his friends have little choice but to go quietly. The Xantha take their captives to an underground lab, where a Tibari scientist, Professor Atrika, is standing in front of a metal pedestal atop which a glowing orb hovers in mid-air. Atrika claims that his mind projector detected three new arrivals in the city, two of whom possessed the finest minds Atrika had ever encountered; Jamie glumly takes himself out of the running. The Doctor questions why a Tibari is working alongside hostile Xantha, and Atrika explains that his vision for the Tibari differs from that of his president; however, he intends to use the mind projector to change that. The Doctor is horrified when Atrika opens the pedestal to reveal a familiar crystal inside; despite the Doctor's earlier assurances, Atrika has constructed his mind projector using Dalek technology.

The Doctor tries to warn Atrika against allying himself with the Daleks, but Atrika is confident of victory; once the Xantha and Tibari are united under his rule, he will go on to build the greatest empire the galaxy has ever seen. Identifying Zoe as the one with the eidetic memory, he explains his plan to her in detail so that her mind will be primed to accept his control. He intends to project her consciousness into a physical avatar generated through quantum entanglement, and use his own latent psychic abilities to force Zoe's avatar to assassinate the Tibari President. Her avatar will be sent to the President's flagship unarmed, and when the time is right, a psychic weapon will be transmitted to her, along with the trigger phrase that will turn her into an unwilling assassin. The Xantha and Tibari will blame each other, and once their forces have been divided and depleted, the Daleks will move in to seize control and build a new empire under Atrika's rule. In the future, Atrika intends to use the projector to undermine other worlds' defences; he intends to splinter the Doctor's mind into 100 simultaneous avatars for that project, very probably killing the Doctor in the process. For now, however, the Daleks want to conduct a simple test to prove that the projector works.

Zoe is both fascinated by the science behind the project and horrified by the use to which it will be put. The Doctor demands to know why Atrika wishes to destroy his people's hope for peace, and Atrika reveals that he fought alongside Ram Vendel during the war, only for his contributions to be forgotten while she was elected President. Atrika then has the Xantha connect Zoe to the mind projector and injects a microscopic control chip into her brain. Zoe feels herself falling out of her body, and awakens aboard a spaceship; her real body is still in the lab, and she's effectively having a physical out-of-body experience. Atrika uses his own psi powers to prevent Zoe's avatar from calling out for help, and, desperate, she lets out a mental cry to the Doctor -- and suddenly feels another presence in the psychic landscape. The Daleks have been monitoring the experiment, and they've just heard Zoe call out to their greatest enemy...

Atrika forces Zoe to walk out into the ship's corridor, where she is caught by two Tibari guards, Erebon and Merka. The guards question and search her, but she isn't carrying a weapon -- yet. Atrika instructs her to tell the guards that she's a human stowaway fleeing from the battle on Barrier Colony 9, and when the guards report her presence to President Vendel, the President orders them to bring Zoe to her. Vendel is fascinated by human beings, as her people have discovered that an unknown race of aliens transplanted their ancestors to their current homeworld from the oceans of Earth; the Tibari and Xantha share a common ancestry. Atrika causes Zoe to suffer pain whenever she tries to warn Vendel of the danger she's in, and Vendel assumes that Zoe's convulsions are due to the exhaustion and starvation she suffered in her flight from Barrier Colony 9. Vendel and Zoe do manage to share an educated conversation about astrophysics before Vendel orders Erebon to take Zoe to a prison cell to rest; she can't be given free run of the ship, but she will be well treated.

Atrika protests that the Daleks have broken cover without consulting him, and the Daleks threaten to exterminate him as well for opposing them. However, the Doctor points out that the Daleks need Atrika alive to operate the mind projector, they need the Doctor alive so they can use his mind to magnify its power, and they need Jamie and Zoe alive so the Doctor will co-operate. The Daleks concede the point, but Atrika has now seen for himself their eagerness to kill; nevertheless, he manages to convince himself that he's still in control, and when the Daleks order him to test the machine on the Doctor, he injects a chip into the Doctor's brain.

The Doctor's avatar appears next to Zoe, and he uses his sonic screwdriver to open the cell door, promising to help Zoe resist Atrika's control. However, the psychic weapon is now fully charged, and Atrika now sends the trigger phrase to Zoe. Zoe bolts out of the cell, and the Doctor tries to stop her but only succeeds in locking himself inside the cell once again. Zoe triple-locks the cell door behind her and sets off, guided through the ship's corridors by Atrika; he has fine-tuned the projector so that nobody on the ship can perceive her avatar as it makes its way to the observation gallery above the peace conference chamber. He tells her in detail how to assassinate President Vendel, simultaneously gloating over his plans and imprinting the instructions in her mind so that she'll be unable to resist his control. Zoe arrives at her destination, and as Vendel and the diplomats enter the chamber and prepare to take the first steps towards peace, Atrika transmits the psychic weapon into Zoe's hand -- a gun that resembles a Dalek weapon. Just as Zoe is about to pull the trigger, however, the Doctor arrives in the gallery and snatches the weapon from her hand. With his help, Zoe manages to shake off Atrika's control.

The Daleks force Atrika to inject them with receptor chips, and their avatars materialise in the gallery; however, rather than kill the Doctor and President Vendel themselves, they order the Doctor to complete the test so they will know whether the projector can be used as a weapon. The Doctor urges Atrika to turn on his so-called allies, but Atrika reveals that the weapons in the Daleks' real bodies are set to fire automatically at any sign of betrayal. The Daleks then reveal that they intend to destroy both the Tibari and Xantha homeworlds once they have control of the mind projector, and use their real bodies to exterminate Atrika's Xanthan allies just to prove their power over him. Furious and horrified, Atrika finally accepts that they had no intention of keeping their side of the bargain, and despite knowing that the Daleks will kill him, he transmits a warning to the Tibari, confessing all.

The Daleks exterminate Atrika for his betrayal and turn on the Doctor and Zoe, deciding to cut their losses and find some other way to test the projector. Before they can do so, however, a black shadow appears in the gallery and engulfs them. Atrika managed to inject a receptor chip into his own body before he was shot, and he has transmitted his consciousness into the Daleks' avatars, controlling them just as he controlled Zoe earlier. His mind won't last long without his body to support it, but he manages to give the Doctor a message for Jamie: the experiment can be aborted by pressing the "extricate" button on the mind projector. Now understanding that the Daleks hate everything that isn't a Dalek, Atrika amplifies the fear within their minds, forcing them to feel the pain and horror of every death they've ever caused. Despite the Daleks' vulnerability, the Doctor refuses to lower himself to their level by killing them, and to save Zoe the same decision, Atrika uses the last of his strength to force the Dalek Commander to exterminate his two underlings and then self-destruct.

Atrika's death lowers the psychic shield hiding the avatars from discovery, but the Doctor manages to call out to Jamie, who presses the button as instructed. The avatars disappear from the gallery just as the Tibari guards burst in, and the Doctor and Zoe wake up back in their own bodies. President Vendel appears on the lab's communications screen, responding to the warning that Atrika had sent, and the Doctor introduces himself and confirms the misguided scientist's warning. Vendel promises to remember that Atrika saw the error of his ways in the end, and to work towards a lasting peace so the Xantha and Tibari will be ready to fight the Daleks should the time come; she also strongly implies that her own latent psi powers enabled her to see the truth in Zoe's mind when they spoke. The Doctor uses his sonic screwdriver to destroy the Dalek crystal at the heart of the mind projector, and he, Jamie, and Zoe depart.