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The crew at the completion of the Farpoint Mission

"Just hoping this isn't the usual way our missions will go, sir."
"Oh no, Number One. I'm sure most will be much more interesting.
– William T. Riker and Jean-Luc Picard, 2364 ("Encounter at Farpoint")

The Farpoint Mission was the first official mission conducted in 2364 by the newly commissioned and launched Federation flagship, the Galaxy-class USS Enterprise-D.

The basics of the mission were explained in a log entry by Captain Jean-Luc Picard:

"Captain's log, Stardate 41153.7. Our destination is planet Deneb IV beyond which lies a great, unexplored mass of the galaxy. My orders are to examine Farpoint, a starbase built there by the inhabitants of that world."

The crew of the Enterprise, which had not yet fully assembled, maintained several opinions on their first mission. Data felt that the mission was easy, as it was to "simply solve the mystery of Farpoint Station." Picard, on the other hand, felt "that Starfleet's orders [were] difficult," as the details of their mission included negotiating "a friendly agreement for Starfleet to use the base, while at the same time snoop around to find out how and why the lifeforms [there] built it." Deanna Troi thought the name of their destination, "Farpoint Station," was in itself "mysterious," whereas Miles O'Brien thought it sounded like "a fairly dull place."

En route to Farpoint, the Enterprise was intercepted by an entity who identified himself as Q. Q put the crew of the Enterprise on trial for the "crimes of Humanity," for which they were found guilty. In a plea for mercy, Picard petitioned the court to accept his crew as proof of what Humanity had become, by requesting that they be tested in their long mission ahead.

Q agreed with Picard's request, and revealed that their "immediate destination offers far more challenge than [they] can possibly imagine," making the Farpoint Mission, in Q's mind, "an excellent test." (TNG: "Encounter at Farpoint")

In the anti-time past, Starfleet canceled the mission to Farpoint and diverted the Enterprise to the Romulan Neutral Zone to investigate a large spatial anomaly that had appeared there in the Devron system. Picard, nevertheless, initially chose to ignore the order and continue to proceed to Deneb IV, but later decided to proceed to the Neutral Zone. In the alternate 2370, Picard asked Troi if she recalled, during the Farpoint Mission, the Enterprise being diverted from Farpoint to the Neutral Zone to investigate a spatial anomaly. She did not. (TNG: "All Good Things...")

In 2371, while Data (who had just had his emotion chip installed) and Geordi La Forge were investigating the attack on the Amargosa observatory, Data began to laugh hysterically at a joke shared between Riker and La Forge during the Farpoint Mission. (Star Trek Generations)

In 2379, at the private memorial held for Data on the USS Enterprise-E, William T. Riker recalled his first meeting with Data on the holodeck during the mission. (Star Trek Nemesis)