The study of Ceresa cycles of Fermat curves has a rich history, going back to Bruno Harris’ fundamental work in the early 80s, where he showed via a Hodge-theoretic argument that the Ceresa cycle of the Fermat curve F(4) of degree 4 is algebraically nontrivial, thereby giving the first explicit example of an algebraic cycle that is homologically trivial but algebraically nontrivial. Soon after, Bloch used an l-adic argument to show that the Ceresa cycle of F(4) is, in fact, of infinite order modulo algebraic equivalence. Since then, Harris’ and Bloch’s approaches have been adapted to other Fermat curves (in particular, by Otsubo, Tadokoro, and Kimura), giving rise to many interesting results. However, despite these efforts, until very recently the nontriviality of Ceresa cycles of Fermat curves modulo rational equivalence (let alone, algebraic equivalence) was not known unconditionally for most prime degrees. The goal of this talk is to discuss some recent developments in this direction. The talk is based on a joint work with Kumar Murty.