When did Fall Out Boy became famous?

Fall Out Boy started gaining recognition and mainstream success around the early to mid-2000s, particularly with the release of their major-label debut album "Take This to Your Grave" in 2003 and its follow-up album "From Under the Cork Tree" in 2005. These releases garnered significant commercial success and helped propel the band to fame, solidifying their status in the alternative rock and emo scenes.