We review measurements performed using scanning tunneling microscopy of the motion of impurity atoms in the Cu(001) surface. Like several other elements, the impurity that we have introduced, In, tends to embed itself in the first atomic layer of this surface. Via the motion of the embedded In atoms, we obtain direct information on the motion of the Cu atoms in the surface. In other words, we employ the In atoms as tracer particles to investigate the intrinsic motion in the first Cu layer. The peculiar statistics of the two-dimensional In diffusion allows us to conclude that the motion is assisted by a rapidly diffusing entity, which we identify as a surface vacancy, i.e. a single missing Cu atom in the outermost Cu layer. A comparison with model calculations of the statistics of the vacancy-assisted motion of terrace atoms shows that there must be an attractive interaction between an embedded In atom and a vacancy, which makes the In atom somewhat more mobile than a Cu surface atom. Such an attraction is indeed found in embedded-atom-method calculations. Nevertheless, the temperature dependence of the indium motion provides an accurate estimation of the sum Eform+Eact, representing the sum of the formation energy of a vacancy and the activation energy for the motion of vacancies through a clean Cu(001) surface.