This paper reviews issues and developments in the field of structure-environment interaction problems, in which the environment is an elastic body, possibly unbounded. It covers in particular the fields of soil-structure interaction, ground-borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium. The general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled, and the domain decomposition technique classically used for its description is put up. Extensions for the cases of nonlinear structure and uncertain environment are then discussed. Finally, the ongoing research in the fields of unbounded interface, moving interface, and multiple interfaces are reviewed and summarized.