The structure of the closed cone of curves on a smooth complex projective variety X may depend on the geometry of X. By the Cone Theorem the closed cone of curves on X is rational polyhedral whenever -K_X is ample, while some abelian surfaces show us examples of non-polyhedral cones. In fact a lot of abelian varieties have non-polyhedral cones. We will determine the condition that the closed cone of curves on an abelian variety is rational polyhedral. I'll closely follow Thomas Bauer's paper with the same title.