We establish a stability result for viscosity solutions of Hamilton-Jacobi equations for a sequence of domains converging in the Hausdorff sense. A background of this study is domain perturbation problems for Hamilton-Jacobi equations. In this talk we introduce the idea of proving the stability of viscosity solutions, starting with the definition of metric viscosity solutions. A key to the proof is a characterization of the metric viscosity solutions by the squared distance functions. This talk is based on a joint work with Atsushi Nakayasu (Kyoto Univ.).