  Let $X$ be a smooth projective variety over the complex numbers. One knows by
the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral
whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample,
however, it is in general difficult to determine the structure of $\bar NE(X)$.
The purpose of this paper is to study the cone of curves of abelian varieties.
Specifically, the abelian varieties $X$ are determined such that the closed
cone $\bar NE(X)$ is rational polyhedral. The result can also be formulated in
terms of the nef cone of $X$ or in terms of the semi-group of effective classes
in the N\'eron-Severi group of $X$.
