For some Action Heroes, a fedora is just the kind of hat to wear while in a Fight Scene, Chase Scene, or just doing anything Badass.
In some of the adventure Film Serials, the fedora would be worn by the hero. When Indiana Jones came along, it solidified the hat as a sign of the adventurer (at least to modern audiences). It can also become an Iconic Item. Another function the fedora serves is [...] [tvtropes.org]