The external surface of a material has an atomic or molecular structure that is different from the bulk material. So does any internal interface within a material. Because of this, the energy of a material or any grain or particle within it increases with the curvature of its bounding surface, as described by the Gibbs-Thomson equation. This chapter explains how surfaces control the nucleation of new phases during reactions such as solidification and precipitation, the coarsening and growth of particles during heat treatment, the equilibrium shape of crystals, and the surface adsorption and segregation of solutes and impurities. The Gibbs-Thomson was predated by a number of related equations; it is not clear whether it is named after J. J. Thomson or William Thomson (Lord Kelvin); and it was not put into its current usual form until after Gibbs’, Thomson’s and Kelvin’s time. J. J. Thomson was the third Cavendish Professor of Physics at Cambridge University. He discovered the electron, which had a profound impact on the world, notably via Thomas Edison’s invention of the light bulb, and subsequent building of the world’s first electricity distribution network. William Thomson was Professor of Natural Philosophy at Glasgow University. He made major scientific developments, notably in thermodynamics, and he helped build the first trans-Atlantic undersea telegraph. Because of his scientific pre-eminence, the absolute unit of temperature, the degree Kelvin, is named after him.