Bulma is there from Dragon Ball's first episode to Daima. She rarely fights, yet in nearly every saga one of her inventions makes the difference. Bulma's profile tells you who she is; here we follow her saga by saga, from the girl hunting for a boyfriend to the mother of Trunks and Bulla.
Dragon Ball
Emperor Pilaf Saga (episodes 1-13)
It all starts in episode 1: Bulma is hunting the Dragon Balls with the radar she invented, hoping to wish for the perfect boyfriend, and in the mountains she meets Goku, who keeps the four-star ball inherited from his grandfather. In the Emperor Pilaf Saga she leads a group that grows to include Master Roshi, Oolong, Yamcha and Puar. The boyfriend wish falls through, though: in episode 12 Oolong beats Pilaf to it and asks Shenron for a pair of panties.
Red Ribbon Army Saga (episodes 29-68)
In the Red Ribbon Army Saga Goku tracks her down in West City with a broken radar. In episode 44 she fixes it, introduces him to her father and decides to come along, shrunk down by another of her inventions. With Krillin she finds the pirates' treasure in episode 52 and later builds a flying camera to find Goku.
From the Fortuneteller Baba Saga to the Piccolo Jr. Saga (episodes 69-153)
In the Fortuneteller Baba Saga she checks the radar and realizes the last ball is not showing up because something or someone has swallowed it. In the Piccolo Jr. Saga she keeps the stones of the Dragon Balls, inert after Shenron's death, and three years later meets her friends again at the new tournament.
Dragon Ball Z
Saiyan and Frieza Sagas (episodes 1-107)
In the Saiyan Saga Bulma modifies Raditz's scouter to read power in numbers Earthlings can understand. In the Frieza Saga she wants to go to Namek to bring her friends back, but in episode 37 she accidentally blows up the Saiyans' ship by pressing the wrong button on the remote. Mr. Popo then shows her the ship Kami came to Earth in: Bulma fixes it with her father and sets off with Gohan and Krillin. On Namek, in a filler, Captain Ginyu, stuck in a frog's body, steals hers, and she only gets it back in episode 91. After Namek she puts Vegeta up at Capsule Corporation.
Androids and Cell Sagas (episodes 118-194)
After splitting with Yamcha, she gets together with Vegeta: in the Androids Saga, at the island where the androids are due, the gang meets baby Trunks, her son with Vegeta, in episode 126. In the alternate future, another Bulma builds the time machine. In the Cell Saga she discovers the second time machine, Cell's, and in episode 148 works on the remote to shut down the androids, based on Dr. Gero's blueprints, which Krillin will end up destroying. With her father she repairs Android 16 and removes his bomb.
Majin Buu Saga (episodes 200-291)
In the Majin Buu Saga she builds the watch that turns Gohan into the Great Saiyaman, then in episode 239 gathers the Dragon Balls to revive Majin Vegeta's victims. Majin Buu turns her into chocolate and eats her; the Namekian Dragon Balls bring her back, and Bulma gives her energy to the final Spirit Bomb.
Dragon Ball GT
Black Star Dragon Ball and Baby Sagas (episodes 1-40)
In the Black Star Dragon Ball Saga she gets the spaceship ready for Goku's trip in episode 2. In the Baby Saga the parasite takes control of her and makes her his queen: Bulma organizes the migration to the new planet and builds the Blutz wave generator that gives Baby his Golden Great Ape form. The Sacred Water frees her.
Shadow Dragon Saga (episodes 48-64)
In the Shadow Dragon Saga she puts the same technology to good use: in episode 55 she explains to Vegeta how he could become a Super Saiyan 4, and in episode 59 she hits him with the portable generator.
Dragon Ball Super
Battle of Gods Saga (episodes 1-18)
The Battle of Gods Saga revolves around her birthday party. When Beerus ruins it, in episode 7 Bulma slaps him, the god hits back and Vegeta explodes. Then she keeps the gods happy with Earth food: in episode 16 she has lunch with Whis, and her instant noodles convince the angel to train Vegeta.
Resurrection 'F' and Universe 6 Sagas (episodes 19-46)
In the Resurrection 'F' Saga Jaco warns her of Frieza's return, and she is the one who contacts Whis to call Goku and Vegeta back. In the Universe 6 Saga she makes a radar for the Super Dragon Balls and in episode 41 tracks down the last one.
Future Trunks Saga (episodes 47-76)
In the Future Trunks Saga Future Bulma is killed by Goku Black in episode 47. The present-day Bulma works on the time machine, follows the Saiyans to the future and shows Trunks a recording of Piccolo's Evil Containment Wave.
Universe Survival Saga (episodes 77-131)
In the Universe Survival Saga she promises a hundred million zeni to every Universe 7 fighter if they win, and in episode 83 Whis speeds up Bulla's birth so that Vegeta can take part.
Dragon Ball Daima
The journey through the Demon Realm (episodes 1-20)
Shrunk like the others (see our Daima episode guide), in episode 2 she is brought to Kami's Lookout with Vegeta to repair Shin's old ship. Out of fuel, she reaches the Demon Realm with Vegeta and Piccolo thanks to Hybis. In episode 13 she repairs the ship with Panzy on a planet of giants, and in the finale reins in an exhausted Vegeta.
Movies
In Dragon Ball Super: Broly we learn she uses the Dragon Balls for small cosmetic touch-ups; in Super Hero she has set up a Capsule Corporation branch that collects the Dragon Balls so they do not fall into the wrong hands.
Our take
Bulma is the thread that ties the series together. Without her radar there would be no first adventure, without her ship no trip to Namek, without Future Bulma's time machine no Trunks.
Then there is her temper. She is the same impulsive girl who slaps a God of Destruction over a ruined party, just a few decades older. The series let her grow up without ever taming her.
For her family, read Vegeta saga by saga and Trunks saga by saga; for her first friend there is Goku's profile.