  We first prove that, unlike the biharmonic case, there exist triharmonic
curves with nonconstant curvature in a suitable Riemannian manifold of
arbitrary dimension. We then give the complete classification of triharmonic
curves in surfaces with constant Gaussian curvature. Next, restricting to
curves in a 3-dimensional Riemannian manifold, we study the family of
triharmonic curves with constant curvature, showing that they are Frenet
helices. In the last part, we give the full classification of triharmonic
Frenet helices in space forms and in Bianchi-Cartan-Vranceanu spaces.
