The wizard who guides Bilbo and Frodo in J.R.R. Tolkien’s Middle-earth is Gandalf.
Gandalf appears in The Hobbit as the organizer of the quest to reclaim the Lonely Mountain. In The Lord of the Rings, he helps Frodo understand the danger of the Ring and supports the Fellowship during its journey. He is known as Gandalf the Grey until his confrontation with the Balrog in Moria, after which he returns as Gandalf the White.
Gandalf is one of the Istari, spiritual beings sent to Middle-earth to oppose Sauron. His original name in Valinor is Olórin. Tolkien’s other named wizards include Saruman, Radagast, and the two mysterious Blue Wizards. Gandalf is therefore not the only wizard in the mythology, but he is the one most directly connected with both Bilbo’s and Frodo’s quests.