The USS Enterprise meets with the USS Fearless to bring aboard Mr. Kosinski (Stanley Kamel), a Starfleet propulsion expert who plans to run tests on the warp engines to improve their efficiency. Along with Kosinski is his assistant, an alien being from Tau Alpha C whose name is unpronounceable by humans, thus called The Traveler (Eric Menyuk). As Kosinski and his assistant explain the tests to the Engineering crew, Wesley Crusher, also invited to watch the tests, quickly grasps what the tests are to accomplish, and the alien shows admiration for Wesley's problem solving abilities. The test is started but quickly goes awry when the Enterprise gains an incredible boost of speed surpassing the known capabilities of warp engines. Captain Picard orders a halt, and once stopped, the crew finds themselves on the far side of the M33 Galaxy, more than 2.7 million light years from the Milky Way, 300 years to get back. Though Kosinski is initially pleased with the results he believed to be from mistaken parameters, he is reprimanded by Picard, and suggests to simply redo the process to return home. Wesley warns Commander Riker that during the warp test, the alien appeared to phase out of reality; when Kosinski starts the second test, both Wesley and Riker observe the alien phasing out, appearing even more tired. The Enterprise again bursts into incredible speeds, and when it stops, the crew cannot determine their position, appearing to have gone "where none have gone before". Picard demands Kosinski get the crew home.
As Kosinski, his assistant, and the Engineering crew determine how to reverse the process, the rest of the crew start to experience lifelike visions of their past, an effect of the strange space around them. After experiencing his own vision of his grandmother, Picard surmises they have arrived at the theoretical Outer Rim, one of the oldest parts of the universe, and uses the Red Alert to wake the crew from their visions. Picard learns from Riker that Kosinski had nothing to do with the warp jumps, but instead it was the result of his alien assistant who has taken ill and moved to Sick Bay. Dr. Crusher cannot determine the alien's biology and is unable to treat him. When Picard arrives, the alien explains he is "a Traveler", able to channel pure thought into reality, and had brought humans to the Outer Rim, which bestows similar effects on anyone within it, to see if humans were ready to experience that, and apologizes for putting the ship in danger. The Traveler, speaking privately to Picard, reveals that he travels looking for prodigies in science such as Wesley, and that Picard should help nurture Wesley. Returning to Engineering, the Traveler asks Wesley to help assist him in returning the Enterprise to known space. As they concentrate and start returning the ship to home, the Traveler completely phases out and disappears, while Wesley is observed to phase out as well. The Enterprise suddenly stops, and the crew is relieved to find themselves at the same location before their first warp jump in Federation space. After the incident, Picard promotes Wesley to an acting ensign on the Enterprise for his performance.