Around 1950 Loewner proved that the area of a torus T^2 is at least \sqr(3)/2 Sys(T)^2 where Sys(T) stands for the systole of T (i.e. the length of the shortest closed noncontractible geodesic). Since then similar isosystolic inequalities have been established for different surfaces. I will introduce Gromov’s concept of Filling Radius and show how it can be used to obtain isosystolic inequalities in higher dimensions.