  It is known that the Schr\"odinger flow on a complex Grassmann manifold is
equivalent to the matrix non-linear Schr\"odinger equation and the Ferapontov
flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation.
In this paper, we give a systematic method to construct integrable geometric
curve flows on Adjoint $U$-orbits from flows in the soliton hierarchy
associated to a compact Lie group $U$. There are natural geometric
bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they
correspond to the order two and three Hamiltonian structures on soliton
equations under our construction. We study the Hamiltonian theory of these
geometric curve flows and also give several explicit examples.
