Fear of Flying stars Dugo, a petite bird who has a uniquely ironic fear of flying. It all starts with a nightmare. Dugo wakes up in his bed, and what follows is a series of events that causes him to begin preparing for winter. To a typical bird, preparing for winter would mean flying south or hunkering down somewhere with an ample food supply. Dugo, on the other hand, decides to go shopping at the local food market.

On his way to the market, he meets fellow birds, Lucy and Dick. Observing the change in the weather, Lucy suggests heading South. Dugo is hesitant, makes an excuse for why he cannot leave, and the two other birds go on their way. Days go by, with all other sensible birds having already vacated the area, Dugo is at home alone. Plunged into utter solitude, Dugo is stuck waiting out the storm inside, changing channels on his TV, and imagining himself with Lucy. As more snow piles up outside, Dugo is met by a robber (a masked squirrel), who clears out all his stocked-up food. Desperate and without supplies, Dugo musters up the courage to “fly” to the South.

Conquering the snowy hilltop, struggling against the storm, Dugo arrives at a nearby airport. He has made it this far. As Dugo settles into his airplane seat, he starts to feel the fear creeping up on him. This seems to last until he is finally in the air, flying. Upon reaching his destination, Dugo is greeted by Lucy on the beach. Relieved that he made it, Lucy asks Dugo if he wants to go for a swim, which consequently begins another tale of woe for Dugo, as it is established that he also has a fear of swimming, too.