Master Roshi is at the heart of the first series, steps aside in Z and, surprisingly, gets back in the ring in Super. Master Roshi's profile covers who he is; here we follow the old hermit arc by arc, from his Jackie Chun disguise to his last Mafuba at the Tournament of Power.
Dragon Ball
Emperor Pilaf Saga (episodes 1-13)
He debuts in episode 3: to thank Goku for rescuing Turtle he gives him the Flying Nimbus, while handing Bulma his Dragon Ball in exchange for a peek. In the Emperor Pilaf Saga he returns when Goku and Chi-Chi ask for help with the fire on Mount Frypan: in episode 8 he puts out the flames with a full-power Kamehameha that takes the mountain and the castle with it.
21st Tournament Saga (episodes 14-28)
In episode 14 he agrees to take Goku as a student, provided he gets a pretty girl in return, and Krillin turns up as well. After eight months of training he enters them in the tournament and, in the 21st Tournament Saga, signs up himself as Jackie Chun to keep the boys from getting big heads. He beats Yamcha and Krillin, then in the final against Goku goes as far as destroying the moon to stop the Great Ape and, in episode 28, wins by a whisker.
Red Ribbon Army and Fortuneteller Baba Sagas (episodes 29-83)
In the Red Ribbon Army Saga he lends Goku his submarine and, in episode 49, easily handles the Red Ribbon commander who shows up at Kame House. In the Fortuneteller Baba Saga the fortuneteller turns out to be his sister, and his nosebleed becomes the weapon that makes the invisible man visible. At the end, in episode 78, he advises Goku to set off and train alone.
22nd Tournament Saga (episodes 84-101)
Jackie Chun is back in the 22nd Tournament Saga. He beats the Wolfman, then faces Tien, student of his old rival the Crane Hermit: in episode 94 he takes the Solar Flare, deflects the Kamehameha the young man has just copied and steps out of the ring on purpose. In episode 95 he explains to Tien that it is time to make way for the young and urges him to choose the right path. When the Crane Hermit threatens Chiaotzu, Roshi blasts him away with a Kamehameha.
King Piccolo Saga (episodes 102-122)
In the King Piccolo Saga he hunts for the Dragon Balls with Tien and Chiaotzu to get ahead of the demon his own master, Mutaito, once sealed away. Knowing he cannot win, in episode 111 he attempts the Mafuba: the seal misses the container and Roshi dies, drained.
Piccolo Jr. Saga (episodes 123-153)
The Dragon Balls, restored by Kami, bring him back in episode 126. In the Piccolo Jr. Saga the anime also shows him young: in episode 129 Goku travels into the past and meets him and the Crane Hermit as Mutaito's students.
Dragon Ball Z
Saiyan Saga (episodes 1-35)
He offers to help Goku against Raditz, but Piccolo dismisses him as too weak. In the Saiyan Saga he follows the fight on television from Kame House and, in episode 24, comforts Bulma over Yamcha's death by reminding her of the Dragon Balls.
Cell Saga (episodes 140-194)
He admits he is no longer the strongest man on Earth, but he is still the best at reading a fight. In the Cell Saga, in episode 180, while everyone thinks Goku is close to winning, he realizes Goku cannot do it and knows it himself.
Majin Buu Saga (episodes 200-291)
At the 25th tournament, in the Majin Buu Saga, he may be the only one to sense that Goten and Trunks are hiding inside Mighty Mask. Sheltering at Kami's Lookout, in episode 260 he is turned into chocolate and eaten by Buu; once revived, he gives his energy to the Spirit Bomb. In GT he shows up in the finale, when Goku stops by to say goodbye.
Dragon Ball Super
Battle of Gods and Resurrection 'F' Sagas (episodes 1-27)
He attends Bulma's party in the Battle of Gods Saga, then in the Resurrection 'F' Saga he is back in action: in episode 21 he brings the senzu beans and takes on Frieza's soldiers with Gohan, Piccolo, Krillin and Tien.
Future Trunks Saga (episodes 47-76)
In the Future Trunks Saga he is the keeper of the Mafuba: in episode 62 he teaches it to Goku so he can seal Zamasu. Sending Krillin and Goku into the Forest of Fear is also his idea, a drastic therapy that in episode 76 gives his student back the will to fight.
Universe Survival Saga (episodes 77-131)
In the Universe Survival Saga Goku recruits him for the Tournament of Power. In episode 89 he falls under Yurin's spell, so he trains with Puar to resist female charm. In the tournament he is a revelation: in episode 105 he single-handedly eliminates Caway, Dercori and Ganos of Universe 4, then collapses and Goku has to revive him. When Frost deflects his Mafuba and traps Vegeta, Roshi frees him with his last strength and leaves the arena. In the manga he is instead eliminated by Jiren, but his lesson helps Goku reach Ultra Instinct.
Did you know? In episode 8 of the anime Chi-Chi refuses to believe this little old man is the famous Roshi, so he shows her his driver's license to prove it.
Our take
Master Roshi has a path unlike anyone else's: he is the strongest at the start and chooses not to be anymore. The Jackie Chun disguises exist to make his students lose at the right time, the forfeit against Tien to hand the future to the young, the Mafuba against King Piccolo to die as a teacher.
Super grants him a comeback that erases none of this: at the Tournament of Power he wins not with strength but with experience, the same experience that in Z let him understand fights better than the people fighting them.
Also revisit the profiles of Goku and Krillin, his most famous students, and Tien, the young man he persuaded to change course. To follow their stories arc by arc too: Goku saga by saga and Krillin saga by saga.