Let (M ,ğ) be an $N$-dimensional smooth (compact or noncompact)
Riemannian manifold. We introduce the elliptic Jacobi-Toda system on (M ,ğ). We review
various recent results on its role in the construction of solutions with multiple interfaces of the Allen-Cahn equation on compact manifolds and entire space, as well as multiple-front traveling waves for its parabolic counterpart.