  The concept of catenary has been recently extended to the sphere and the
hyperbolic plane by the second author [L\'opez, arXiv:2208.13694]. In this
work, we define catenaries on any Riemannian surface. A catenary on a surface
is a critical point of the potential functional, where we calculate the
potential with the intrinsic distance to a fixed reference geodesic. Adopting
semi-geodesic coordinates around the reference geodesic, we characterize
catenaries using their curvature. Finally, after revisiting the space-form
catenaries, we consider surfaces of revolution (where a Clairaut relation is
established), ruled surfaces, and the Gru\v{s}in plane.
