The search for an ultimate axiomatization of mathematics is inevitably incomplete. However, this does not preclude the possibility of strong and natural extensions to the standard axioms of set theory (ZFC) which shed light on many unresolved mathematical questions. Gödel suggested searching for strong axioms of infinity (known as large cardinals) as potential candidates for such extensions -- a pursuit that has continued for over half a century. This talk will survey several central developments in this search, including results concerning finite graphs, questions about Lebesgue measurability of projective sets of reals, the inner model program, and current efforts toward identifying an "ultimate" model of set theory via canonical inner models.