A general thermodynamics of surfaces between elastic media is developed which extends an earlier treatment by Andreev and Kosevich to embody the Gibbsian description of fluid-fluid interfaces. Complementary to the well known surface stress the concept of surface strain is introduced. The thermodynamic variables of the surface energy and the corresponding Gibbs-Duhem- and Maxwell-relations are investigated. We study two examples in detail:a) a surface between solids that cannot transform into each other, andb) an interface between a solid and its melt. In the latter case we show that the surface tension only depends on the tangential strain.