Around 1949, C. Loewner proved the first systolic inequality. Namely, he showed that the area of every Riemannian 2-torus with unit systole (that is, the length of the shortest noncontractible loop equals 1) is at least that of the equilateral flat torus. We will talk about this inequality and show that it still holds for hyperelliptic surfaces. This is joint work with M. Katz.