We will talk about the Ceresa class, which is the image under a cycle class map of a canonical homologically trivial algebraic cycle associated to a curve in its Jacobian. In his 1983 thesis, Ceresa showed that the generic curve of genus at least 3 has nonvanishing Ceresa cycle modulo algebraic equivalence. The Ceresa class vanishes for all hyperelliptic curves and was expected to be nonvanishing for non-hyperelliptic curves. Strategies for proving Fermat curves have infinite order Ceresa cycles due to B.Harris, Bloch, Bertolini-Darmon-Prasanna, Eskandari-Murty use a variety of ideas ranging from computation of explicit iterated period integrals, special values of p-adic L functions and points of infinite order on the Jacobian of Fermat curves. We will survey several recent results about the Ceresa class.