One of the most famous Star Trek episodes, this one features not only the fecund furball species in the title, but also the Klingons, who would become one of the most enduring alien species in the franchise.
Probably the most inexpensive alien creature ever made for the show, a tribble is a purring ball of fur. The trouble with them? They reproduce at an alarming rate. Indeed, as McCoy points out, they're born pregnant. The crew gets a furry lesson in logarithmic math as the creatures start to fill up the Enterprise.
According to David Gerrold, who wrote the script, the episode was meant to be a commentary on the dangers of invasive species.
"The Trouble with Tribbles" was revisited by Star Trek: Deep Space Nine, when the DS9 crew travels back in time to witness the events in the episode.
[Editor's note: An earlier version said that this episode introduced the Klingons. The Klingons actually made their first appearance in the first-season episode "Errand of Mercy." The Monitor regrets the error.]