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Liquid drops on soft solids generate strong deformations below the contactline, resulting from a balance of capillary and elastic forces. The movement ofthese drops may cause strong, potentially singular dissipation in the softsolid. Here we show that a drop on a soft substrate moves by surfing a ridge:the initially flat solid surface is deformed into a sharp ridge whoseorientation angle depends on the contact line velocity. We measure this anglefor water on a silicone gel and develop a theory based on the substraterheology. We quantitatively recover the dynamic contact angle and provide amechanism for stick-slip motion when a drop is forced strongly: the contactline depins and slides down the wetting ridge, forming a new one after atransient. We anticipate that our theory will have implications in problemssuch as self-organization of cell tissues or the design of capillarity-basedmicrorheometers.

Droplets move over viscoelastic substrates by surfing a ridge