The Students of Baldesion is an organization of scholars in Final Fantasy XIV, based near Sharlayan on the Isle of Val. The organization exists to study arcane mysteries across the world, and recently have focused on the echo, Ascians, and the Umbral Calamity phenomenon. They are close allies with the Scions of the Seventh Dawn.
Final Fantasy XIV: A Realm Reborn
They are frequently mentioned during the A Realm Reborn main storyline as contacts for Urianger Augurelt and Minfilia Warde. Later, during the Crystal Tower storyline, one of their members, G'raha Tia, is revealed to have been recruited by Cid and Rammbroes Zasertylsyn for the investigation.
Eventually the Scions lose contact with the Students of Baldesion. Urianger discovered that something happened to the Isle of Val, as though some great and powerful magic similar to Ultima was used and the island was "utterly....gone."
Final Fantasy XIV: Heavensward
Krile, grandaughter of the organization's leader Galuf and the only known survivor of the Isle of Val's disappearance, arrived in Eorzea to help the Scions. She had been saved from the island's disappearance by the intervention of Hydaelyn.
Final Fantasy XIV: Stormblood
The Isle of Val was discovered on the other side of the world from where it should be, in the ocean near Kugane. Upon arriving on the deserted island and speaking with Krile, the Warrior of Light learned that they had given the island the new name "Eureka". Krile's goal was to both learn what happened and to see if any surviving members of the Students of Baldesion could be found on the island. Ejika Tsunjika is also revealed to have survived the mysterious incident.
Over the course of the investigation, it is discovered that the incident that teleported the Isle of Val away and killed everyone on it was intentionally done by Galuf himself. He plunged the entire island in the aetherial sea in order to kill the Ascian Emmerololth and keep them from reaching the primal Eureka.
Final Fantasy XIV: Endwalker
Krile begins working to restore the Students of Baldesion while in Sharlayan, and the Scions of the Seventh Dawn are permitted access to Sharlayan under the guise of helping her. After the Final Days are averted, G'raha Tia joins Krile's efforts to restore the Students.
"Baldesion" comes from Krile Mayer Baldesion and her grandfather, Galuf Halm Baldesion from Final Fantasy V. Their base the Isle of Val is named after Bal Castle from the same game; Val and Bal are both spelled the same in Japanese.