Hydrodynamic couplings of elastic filaments with external flows are crucial for the functioning of various biological and synthetic processes, such as the locomotion of flagellated and ciliated organisms. In this talk, we will analyze the steady-state deformation of elastic filaments subject to external forces and flows, focusing on the regime of low Reynolds numbers. We will begin by discussing the basic equations for the problem, which consist of the nonlinear Euler-Bernoulli equations coupled with a local slender-body description of the hydrodynamic forces. In dimensionless form, the problem depends on a single compliance parameter. By employing singular perturbation techniques, we will derive closed-form approximations in the limits of small and large compliances. For a filament subject to an external force (e.g., gravity), we will also show the existence of multiple branches of solutions, associated with distinct terminal shapes and speeds.  Our theoretical predictions and asymptotic approximations are in good agreement with experimental observations and numerical solutions of the full elastohydrodynamic problem.