Abstract We show that every smooth cubic hypersurface $$X$$ in $$\mathbb P^{n+1}$$ , $$n\ge 2$$ , is algebraically elliptic in Gromov’s sense. This gives the first examples of nonrational projective manifolds elliptic in Gromov’s sense. We also deduce that the punctured affine cone over $$X$$ is elliptic.