In this paper we use a lubrication approximation to investigate the locally unidirectional gravity-driven draining of a thin rivulet of Newtonian fluid down a slowly varying substrate. The work generalizes the recent study by Duffy and Moffatt [Chem. Eng. J.60, 141 (1995)] of gravity-driven draining down a locally planar substrate to include the effects of substrate variation transverse to the direction of flow. Asymptotic and numerical results are obtained for several simple convex and concave transverse substrate profiles. In all the cases investigated these results show a number of common features. In particular, they show that a single stable slowly varying rivulet running continuously from the top to the bottom of a large horizontal circular cylinder is possible only if the transverse substrate profile is a sufficiently shallow trough. If the profile is a deeper trough then no such rivulet is possible near the bottom of the cylinder, while if the profile is a ridge then no such rivulet is possible near the top of the cylinder. ©1998 American Institute of Physics.