Summary: If a droplet smaller than the capillary length is placed on a substrate with a conical shape, it spreads by itself in the direction of growing fibre radius. We describe this capillary spreading dynamics by developing a lubrication flow approximation on a cone and by using the perturbation method of matched asymptotic expansions. Our results show that the droplet appears to adopt a quasi-static shape and the predictions of the droplet shape and the spreading velocity from the two mathematical models are in excellent agreement. At the contact line regions, a large pressure gradient is generated by the mismatch between the equilibrium contact angle and the apparent contact angle that maintains the viscous flow. It is the conical shape of the substrate that breaks the front/rear droplet symmetry in terms of the apparent contact angle, which is larger at the thicker part of the cone than at its thinner part. Consequently, the droplet is predicted to move from the cone tip to its base, consistent with experimental observations.