This talk is about the relationship between the p-rank of a curve and p-ranks iof the Prym varieties of its cyclic covers in characteristic p>0. Prym variety is a central object of study in arithmetic geometry like Jacobian variety. The goal of the talk is to understand various existence results about Prym varieties such as when g>2, Prym varieties of all unramified cyclic degree ell covers of a generic curve of genus g and p-rank f are ordinary. This is joint work with Rachel Pries