The elastic flow, which is the $L^2$-gradient flow of the elastic energy, has
several applications in geometry and elasticity theory. We present stable
discretizations for the elastic flow in two-dimensional Riemannian manifolds
that are conformally flat, i.e.\ conformally equivalent to the Euclidean space.
Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane
as well as any conformal parameterization of a two-dimensional manifold in
${\mathbb R}^d$, $d\geq 3$. Numerical results show the robustness of the
method, as well as quadratic convergence with respect to the space
discretization.