The Prym varieties introduced by Mumford in the 70's opened new perspectives in the study of the moduli space of abelian varieties: Prym varieties are abelian varieties arising from covers between curves. They generalize Jacobians and provide a larger class of abelian varieties which can be understood in terms of 1-dimensional geometric objects. For instance, it is known that the general principally polarized abelian variety of dimension less than or equal to 5 is the Prym variety of some unramified double cover. The Prym map between the moduli space of covers and the moduli space of abelian varieties assigns to each covering its Prym variety. In this talk I will discuss the geometry of the fibres of the Prym map for double covers in low genera, the global injectivity of the Prym map for ramified double covers, as well as the most recent results on the general injectivity of the Prym map for cyclic covers.